Anais do XIV ENAMA

Anais do XIV ENAMA

Home web: http://www.enama.org

Comissao Organizadora

Alannio Barbosa Nobrega (UFCG)

Aldo Trajano Louredo (UEPB)

Angelo Roncalli Furtado de Holanda (UFCG)

Claudianor Oliveira Alves (UFCG)

Denilson da Silva Pereira (UFCG)

Francisco Siberio Bezerra Albuquerque (UEPB)

Gustavo da Silva Araujo (UEPB)

Jefferson Abrantes dos Santos (UFCG)

Jose Lindomberg Possiano Barreiro (UFCG)

Luciana Roze de Freitas (UEPB)

Marcelo Carvalho Ferreira (UFCG)

Pammella Queiroz de Souza (UFCG)

Severino Horacio da Silva (UFCG)

Realizacao: UEPB e UFCG

Apoio:

O ENAMA e um encontro cientıfico anual com proposito de criar um forum de debates entre alunos, professores

e pesquisadores de instituicoes de ensino e pesquisa, tendo como areas de interesse: Analise Funcional, Analise

Numerica, Equacoes Diferenciais Parciais, Ordinarias e Funcionais.

Home web: http://www.enama.org

O XIV ENAMA e uma realizacao conjunta do Departamento de Matematica da Universidade Estadual da

Paraıba e da Unidade Academica de Matematica da Universidade Federal de Campina Grande. O evento estava

previsto para ocorrer em novembro de 2020, porem, considerando a situacao sanitaria causada pela pandemia da

COVID-19, foi adiado para novembro de 2021. Tendo em vista as incertezas quanto ao fim da pandemia no Brasil,

o XIV ENAMA sera realizado de forma totalmente remota no perıodo de 03 a 05 de novembro de 2021.

Os organizadores do XIV ENAMA expressam sua gratidao aos orgaos e instituicoes, DM - UEPB e UAMat -

UFCG, que apoiaram e tornaram possıvel a realizacao do XIV ENAMA.

Comissao Cientıfica

Ademir Pastor (UNICAMP)

Alexandre Madureira (LNCC)

Giovany Malcher Figueiredo (UnB)

Jaqueline G. Mesquita (UnB)

Juan A. Soriano (UEM)

Marcos T. Oliveira Pimenta (UNESP)

Vinıcius Vieira Favaro (UFU)

Comite Nacional

Haroldo Clark (UFDPar)

Sandra Malta (LNCC)

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ENAMA 2021

ANAIS DO XIV ENAMA

03 a 05 de Novembro 2021

ConteudoOn positive solutions of elliptic equations with oscillating nolinearity in RN , por

Francisco J. S. A. Correa, Romildo N. de Lima & Alannio B. Nobrega . . . . . . . . . . . . . . . . . . . . . . . . . 11

Existence and approximation of solutions for a class of degenerate elliptic equations

with Neumann boundary condition, por Albo Carlos Cavalheiro . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

On a precise scaling to caffarelli-kohn-nirenberg inequality, por Aldo Bazan & Wladimir

Neves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Finsler double phase problems involving critical Sobolev nonlinearities, por Csaba

Farkas, Alessio Fiscella & Patrick Winkert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Schrodinger equations with vanishing potentials involving Brezis-Kamin type problems,

por J. A. Cardoso, P. Cerda, D. S. Pereira & P. Ubilla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Problemas do tipo henon com o operador 1-laplaciano, por Anderson dos S. Gonzaga &

Marcos T. O. Pimenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

On a class of elliptic systems of the hardy-kirchhoff type in RN , por Augusto C. R. Costa,

Olimpio H. Miyagaki & Fabio R. Pereira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

A Hessian-dependent functional with free boundaries and applications to mean-field

games, por Julio C. Correa & Edgard A. Pimentel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Positive solutions for a class of fractional Choquard equation in exterior domain, por

Cesar T. Ledesma & Olimpio H. Miyagaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Choquard equations via nonlinear Rayleigh quotient for concave-convex nonlineari-

ties, por Claudiney Goulart, Marcos L. M. Carvalho & Edcarlos D. Silva . . . . . . . . . . . . . . . . . . . . . . 29

Sobre a camada de transicao interna de problemas semilineares nao-homogeneos: a

localizacao da interface, por Sonego, Maicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Elliptic systems involving Schrodinger operators with vanishing potentials., por

Denilson da S. Pereira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Ground states for fractional linear coupled systems via profile decomposition, por

J. C. de Albuquerque, Diego Ferraz & Edcarlos D. Silva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Regularidade de interior para solucoes de equacoes fracionarias que degeneram com

o gradiente, por Disson dos Prazeres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4

Superlinear fractional elliptic problems via the nonlinear Rayleigh quotient with two

parameters, por Edcarlos D. Silva, M. L. M. Carvalho, M. L. Silva & C. Goulart . . . . . . . . . . . . . . . 39

Existence of solutions for a fractional choquard–type equation in R with critical

exponential growth, por Eudes Mendes Barboza, Rodrigo Clemente & Jose Carlos de Albuquerque 41

Critical Metrics of the Sk Operator, por Flavio A. Lemos & Ezequiel R. Barbosa . . . . . . . . . . 43

Existence of solution for implicit elliptic equations involving the p-laplace operator,

por Gabriel Rodriguez V., Eugenio Cabanillas L., Willy Barahona M., Luis Macha C. & Vıctor Carrera

B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Um sistema de tipo schrodinger-born-infeld, por Gaetano Siciliano . . . . . . . . . . . . . . . . . . . . . . . 47

equacoes de schrodinger quaselineares com potenciais singulares e se anulando envol-

vendo nao linearidades com crescimento crıtico exponencial, por Gilson M. de Carvalho,

Yane L. R. Araujo & Rodrigo G. Clemente . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

The limiting behavior of global minimizers in non-reflexive Orlicz-Sobolev spaces, por

Grey Ercole, Giovany M. Figueiredo, Viviane M. Magalhaes & Gilberto A. Pereira . . . . . . . . . . . . . . . 51

Variational free transmission problems of Bernoulli type, por Harish Shrivastava & Diego

Moreira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Compactness within the space of complete, constant Q-curvature metrics on the sphere

with isolated singularities, por Joao Henrique Andrade, Joao Marcos do O & Jesse Ratzkin . . . . 55

Geometric gradient estimates for nonlinear PDEs with unbalanced degeneracy, por

Joao Vitor da Silva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

um problema anisotropico envolvendo o operador 1-laplaciano com pesos ilimitados, por

Juan C. Ortiz Chata, Marcos T. de Oliveira Pimenta & Sergio S. de Leon . . . . . . . . . . . . . . . . . . . . . . 59

Compact embedding theorems and a Lions’ type Lemma for fractional Orlicz–Sobolev

spaces, por Marcos L. M. Carvalho, Edcarlos D. Silva, J. C. de Albuquerque & S. Bahrouni . . . . . . . . 61

coupled and uncoupled sign-changing spikes of singularly perturbed elliptic systems,

por Mayra Soares & Monica Clapp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

On an Ambrosetti-Prodi type problem in RN , por Claudianor O. Alves, Romildo N. de Lima &

Alannio B. Nobrega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

An elliptic system with measurable coefficients and singular nonlinearities, por Lucio

Boccardo, Stefano Buccheri & Carlos Alberto Pereira dos Santos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Fourth-Order Nonlocal type elliptic problems with indefinite nonlinearities, por

Edcarlos D. da Silva, Thiago R. Cavalcante & J.C. de Albuquerque . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

On the frictional contact problem of p(x)-Kirchhoff Type, por Willy Barahona M., Eugenio

Cabanillas L., Rocıo De La Cruz M., Jesus Luque R. & Heron Morales M. . . . . . . . . . . . . . . . . . . . . . . 71

Global multiplicity of solutions for a modified elliptic problem with singular terms,

por Jiazheng Zhou, Carlos Alberto P. dos Santos & Minbo Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

The IVP for the evolution equation of wave fronts in chemical reactions in low-

regularity Sobolev spaces, por Alysson Cunha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Existencia de solucoes periodicas em escoamentos de ferrofluidos, por Jauber C. Oliveira . 77

5

Exact boundary controllability for the wave equation in moving boundary domains

witha star-shaped hole, por Ruikson S. O. Nunes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Maximal attractors for semigroups, por Matheus C. Bortolan . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Controllability under positive constraints for quasilinear parabolic PDEs, por

Miguel R. Nunez Chavez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

on a variational inequality for a beam equation with internal damping and source

terms, por Geraldo M. de Araujo & Ducival C. Pereira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

The nonlinear Quadratic Interactions of the Schrodinger type on the half-line, por

Isnaldo Isaac Barbosa & Marcio Cavalcante . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Vibrations of a bar submitted to an impact, por M. Milla Miranda, L. A. Medeiros & A. T.

Louredo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

About polynomial stability for the porous-elastic system with Fourier’s law, por

Anderson Ramos, Dilberto Almeida Junior & Mirelson Freitas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Existence and exponential decay for wave equation in whole hyperbolic space, por P. C.

Carriao, O. H. Miyagaki & A. Vicente . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Realizability of the rapid distortion theory spectrum, por Ailin Ruiz de Zarate Fabregas,

Nelson Luis Dias & Daniel G. Alfaro Vigo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Stability of periodic solutions of the navier-stokes equations, por Enrique Fernandez-Cara,

Felipe Wergete Cruz & Marko A. Rojas-Medar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

A damped nonlinear hyperbolic equation with nonlinear strain term, por Eugenio Cabanillas L.,

Zacarias Huaringa S., Juan B. Bernui B. & Benigno Godoy T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

comportamento assintotico para as equacoes magneto-micropolares, por Felipe W. Cruz,

Cilon Perusato, Marko Rojas-Medar & Paulo Zingano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

resultados de existencia global para solucoes de equacoes de adveccao-difusao, por

Janaina P. Zingano, Juliana S. Ziebell, Lineia Schutz & Patrıcia L. Guidolin . . . . . . . . . . . . . . . . . . . . 103

Long-time dynamics for a fractional piezoelectric system with magnetic effects and

Fourier’s law, por Mirelson Freitas, Anderson Ramos, Dilberto Almeida Junior & Ahmet Ozer . . . . 105

Global solutions to the non-local Navier-Stokes equations, por Joelma Azevedo, Juan

Carlos Pozo & Santiago, Chile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

existencia global e nao global de solucoes para uma equacao do calor com coeficientes

degenerados, por Ricardo Castillo, Omar Guzman-Rea & Maria Zegarra . . . . . . . . . . . . . . . . . . . . . 109

Existencia de escoamentos de fluıdos magneticos periodicos no tempo, por

Maria Nilde F. Barreto & Mjauber C. Oliveira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Asymptotic behavior of the coupled Klein-Gordon-Schrodinger systems on compact

manifolds, por Cesar A. Bortot, Thales M. Souza & Janaina P. Zanchetta . . . . . . . . . . . . . . . . . . . . . 113

Global regularity for a 1d supercritical transport equation, por Valter V. C. Moitinho

& Lucas C. F. Ferreira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A Kato type exponent for a class of semilinear evolution equations with time-

dependent damping, por Wanderley Nunes do Nascimento, Marcelo Rempel Ebert & Jorge Marques117

6

Existence and continuous dependence of the local solution of non homogeneous KdV-

K-S equation, por Yolanda Santiago Ayala & Santiago Rojas Romero . . . . . . . . . . . . . . . . . . . . . . . . 119

Grothendieck-type subsets of Banach lattices, por Pablo Galindo & Vinıcius C. C. Miranda . 121

Lower bounds for the constants in the real multipolynomial Bohnenblust-Hille

inequality, por Thiago Velanga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Tightness in Banach spaces with transfinite basis, por Alejandra C. Caceres Rigo . . . . . . . . . . 125

results on the frechet space HL(BE), por Luiza A. Moraes & Alex F. Pereira . . . . . . . . . . . . . . . 127

sobre uma reformulacao da hipotese de riemann no espaco de hardy do cırculo unitario,

por Charles F. dos Santos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Teoremas do tipo Banach-Stone para algebras de germes holomorfos em espacos de

Banach, por Daniela M. Vieira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Operadores multilineares somantes e classes de sequencias, por Geraldo Botelho &

Davidson Freitas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Propriedade de Schur polinomial positiva, por Geraldo Botelho & Jose Lucas Pereira Luiz . . . . 135

A semigroup related to the Riemann hypothesis, por Juan C. Manzur . . . . . . . . . . . . . . . . . . . . 137

Asymptotic behavior of solutions to Nonlinear Integral Equations via Renormaliza-

tion, por Gastao A. Braga, Jussara M. Moreira & Camila F. Souza . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Sharp estimates for the covering numbers of the Weierstrass fractal kernel, por

Karina N. Gonzalez, Douglas Azevedo & Thais Jordao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Dirichlet series with maximal bohr’s strip, por Thiago R. Alves, Leonardo S. Brito & Daniel

Carando . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Extensoes de Arens de multimorfismos em espacos de Riesz e reticulados de Banach, por

Geraldo Botelho & Luis A. Garcia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Universal Toeplitz operators on the Hardy space over the polydisk, por Marcos S. Ferreira147

Ciclicidade e hiperciclicidade de operadores de composicao no espaco de Hardy do semi-

plano direito, por Osmar R. Severiano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

aneis linearmente topologizados estritamente minimais, por Patricia C. G. Mauro &

Dinamerico P. Pombo Jr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

The Riemann-Liouville fractional integral as a semigroup in Bochner-Lebesgue spaces,

por Paulo M. Carvalho Neto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

A Hiper-Transformada de Borel Polinomial, por Geraldo Botelho & Raquel Wood . . . . . . . . . 155

some properties of almost summing operators, por Renato Macedo & Joedson Santos . . . . . . . 157

On the Bishop-Phelps-Bollobas theorem for bilinear forms for function module spaces,

por Thiago Grando . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Em direcao a um teorema espectral para semigrupos convolutos, por Aldo Pereira . . . . . . . 161

solutions for functional volterra–stieltjes integral equations, por Anna Carolina Lafeta . 163

A conjectura de Besse, espaco vacuo estatico e espaco σ2-singular, por Maria Andrade . . . 165

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Eigenvalue problems for Fredholm operators with set-valued perturbations, por

Pierluigi Benevieri & Antonio Iannizzotto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

on a class of abel differential equations of third kind, por Samuel Nascimento Candido . . . 169

Convergence of alevel-set algorithm for scalar conservation laws, por Anibal. Coronel . . 171

Lagrangian–Eulerian scheme for general balance laws, por Eduardo Abreu, Eduardo P. Barros

& Wanderson Lambert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

A positive Lagrangian-Eulerian scheme for hyperbolic systems, por Eduardo Abreu,

Jean Francois, Wanderson Lambert & John Perez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

A convergent finite difference method for a type of nonlinear fractional advection-

diffusion equation, por Jocemar de Q. Chagas, Giuliano G. La Guardia & Ervin K. Lenzi . . . . . . . 177

Solution of linear radiative transfer equation in hollow sphere by diamond difference

discrete ordinates and Adomian methods, por Marcelo Schramm, Cibele A. Ladeia & Julio C.

L. Fernandes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

analysis results on an arbitrary-order SIR model constructed with Mittag-Leffler

distribution, por Noemi Zeraick Monteiro & Sandro Rodrigues Mazorche . . . . . . . . . . . . . . . . . . . . . 181

Numerical Analysis for a Thermoelastic Diffusion Problem in Moving Boundary, por

Rodrigo L. R. Madureira & Mauro A. Rincon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

numerical analysis and travelling wave solutions for an internal wave system, por

Willian C. Lesinhovski & Ailin Ruiz de Zarate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Improved Regularity for nonlocal elliptic equations through asymptotic profiles, por

Aelson O. Sobral & Disson dos Prazeres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Existence and nonexistence of solution for a class of quasilinear Schrodinger

equations with critical growth, por Diogo de S. Germano & Uberlandio B. Severeo . . . . . . . . . 189

On the fractional p-Laplacian Choquard logarithmic equation with exponential

critical growth: existence and multiplicity, por Eduardo de S. Boer & Olımpio H. Miyagaki 191

Equacao de Choquard: existencia de solucoes de energia mınima para uma

classe de problemas nao locais envolvendo potenciais limitados ou ilimitados, por

Eduardo Dias Lima & Edcarlos Domingos da Silva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Fully nonlinear singularly perturbed models with non-homogeneous degeneracy, por

Elzon C. Junior, v & Joao Vitor da Silva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

A geometric approach to infinity Laplacian with singular absorptions, por Ginaldo S. Sa

& Damiao J. Araujo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

existencia de solucoes positivas para o p-laplaciano fracionario envolvendo nao

linearidade concavo convexa, por Jefferson Luıs Arruda Oliveira & Edcarlos Domingos da Silva 199

Stabilized hybrid finite element methods for the Helmholtz problem, por Martha H. T. Sanchez

& Abimael F. D. Loula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

problema quaselinear de autovalor com nao-linearidade descontınua, por J. Abrantes

Santos, Pedro Fellype S. Pontes & Sergio Henrique M. Soares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

8

Multiplicity of solutions to a Schrodinger problem with square diffusion term, por

Carlos Alberto Santos, Kaye Silva & Steffanio Moreno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

o problema de dirichlet para uma classe de equacoes do tipo p-laplaciano, por

Glece Valerio Kerchiner & William S. de Matos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

existencia e regularidade para a solucao de um sistema multifasico da eletrohidro-

dinamica, por Andre F. Pereira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Two-dimensional incompressible micropolar fluids model with singular initial data, por

Cleyton N. L. de C. Cunha, Alexis Bejar-Lopez & Juan Soler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Estabilizacao na fronteira nao linear de um sistema termoelastico, por Eiji Renan Takahashi

& Juan Amadeo Soriano Palomino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

An inverse problem for a SIR reaction-diffusion model, por Anibal Coronel &

Fernando Huancas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Solvability of the Fractional Hyperbolic Keller-Segel System, por Gerardo Huaroto &

Wladimir Neves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Controllability of phase-field system with one control, por Sousa-Neto, G. R. & Gonzalez-

Burgos, M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

controlabilidade exata para a equacao kdv via estrategia stackelberg-nash, por

Islanita C. A. Albuquerque, Fagner D. Araruna & Maurıcio C. Santos . . . . . . . . . . . . . . . . . . . . . . . . . 221

Exponential attractor for a class of non local evolution equations, por

Jandeilson S. Da Silva, Severino H. Da Silva & Aldo T. Louredo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Global solutions to the non-local Navier-Stokes equations, por Joelma Azevedo, Juan

Carlos Pozo & Arlucio Viana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Controlabilidade global do sistema de Boussinesq com condicoes de fronteira do tipo

Navier, por F. W. Chaves-Silva, E. Fernandez-Cara, K. Le Balc’h, J. L. F. Machado & D. A. Souza . 227

hierarchical exact controllability of semilinear parabolic equations with distributed

and boundary controls, por F. D. Araruna, E. Fernandez-Cara & L. C. da Silva . . . . . . . . . . . . . . 229

Interacao entre Dissipacao Fracionaria e Nao-Linearidade de Memoria na Existencia

de Solucoes para Equacoes de Tipo Placa, por Luis Gustavo Longen . . . . . . . . . . . . . . . . . . . . . 231

Exponentes crıticos para um sistema parabolico acoplado com coeficientes

degenerados, por Ricardo Castillo, Omar Guzman-rea, Miguel Loayza & Marıa Zegarra . . . . . . . . . 233

sistema de bresse com dissipacao nao-linear na fronteira, por Patricia Vilar Vitor Salinas &

Juan Amadeo Soriano Palomino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Sobre a controlabilidade uniforme dos sistemas Burgers-α nao-viscoso e viscoso, por

Raul K. C. Araujo, Enrique Fernandez-Cara & Diego Araujo de Souza . . . . . . . . . . . . . . . . . . . . . . . . . 237

sistema de bresse com acoplamento termoelastico no momento fletor e lei de fourier,

por Romario Tomilhero Frias & Michele de Oliveira Alves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

lineability of multilinear summing operators, por Lindines Coleta . . . . . . . . . . . . . . . . . . . . . . . 241

polinomios homogeneos nao analıticos e uma aplicacao as series de dirichlet, por

Mikaela A. Oliveira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

9

ideais injetivos de polinomios homogeneos entre espacos de banach, por Geraldo Botelho &

Pedro C. Bazan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Existence of positive solutions for boundary value problems with p-laplacian operator,

por Francisco J. Torres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Esquemas de diferencas finitas para secao circular, por Tatiana Danelon de Assis & Sandro

Rodrigues Mazorche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

ENAMA - Encontro Nacional de Analise Matematica e AplicacoesUEPB - Universidade Estadual da Paraıba eUFCG - Universidade Federal de Campina GrandeXIV ENAMA - Novembro 2021 11–12

ON POSITIVE SOLUTIONS OF ELLIPTIC EQUATIONS WITH OSCILLATING NOLINEARITY IN

RN

FRANCISCO J. S. A. CORREA1, ROMILDO N. DE LIMA2 & ALANNIO B. NOBREGA3

1UAMat, UFCG, PB, Brasil, [email protected],2UAMat, UFCG, PB, Brasil,[email protected],3UAMat, UFCG, PB, Brasil,[email protected]

Abstract

In this paper, we study results of existence and multiplicity of positive solutions for the following semilinear

problem −∆u = λP (x)f(u), in RN

lim|x|→∞ u(x) = 0,

where P ∈ C(RN ,R) and f ∈ C([0,∞),R) is an oscillating nonlinearity satisfying a sort of area condition. The

main tools used are variational methods and sub-supersolution method.

1 Introduction

In this work, we study the existence and multiplicity positive solutions for the problem−∆u = λP (x)f(u) inRN

lim|x|→∞ u(x) = 0,(P )

where f : [0,+∞) → RN is a continuous function satisfy:

(f1) f(0) ≥ 0

(f2) There exist 2m− 1 zeros of f , 0 < a1 < b1 < a2 < b2 < · · · < bm−1 < am such that for k = 1, · · · ,m− 1f(t) ≥ 0, t ∈ (bk, ak+1)

f(t) ≤ 0, t ∈ (ak, bk);

(f3)∫ ak+1

akf(s)ds > 0, for all k ∈ 1, 2, · · · ,m− 1.

Related to P we assume that it is a C+rad(RN ,RN ) function and

(P1)∫RN |x|2−NP (x)dx < +∞;

(P2) P ∈ L1(RN ) ∩ L∞(RN )

and

(P3)∫RN

P (y)|x−y|N−2 dy ≤ C

|x|N−2 , for all x ∈ RN \ 0, for some C > 0.

The existence and multiplicity of solutionts to elliptic problems like (P) in bounded domains with oscillating

nonlinearities, as in (f2), and area condition, like (f3), have been vastly studied since the appearance of the

pioneering papers by Brown and Budin [1, 2]. In [5], Hess improves the aforementioned Brown and Budin’s result,

thanks to minimization arguments and Leray-Schauder degree theory. After, in [4], De Figueiredo , using variational

techniques showed existence of multiple ordered solutions.

Based in the references aforementioned and in the pappers due to Loc and Schmitt [6], Correa, Carvalho,

Goncalves and Silva [3], we use Variational Methods and Comparison Principles to study the existence and

multiplicity of solutions to (P ) in whole RN . We would like to point out that there are some particularities

in the fact that we are working in unbounded domains, some these problems can be overcome using the Riesz

Potential Theory to solutions of (P ).

11

12

2 Main Results

Our main result are the following:

Firstly, we study the existence and multiplicity to problem (P )

Theorem 2.1. Assume that the function f satisfies (f1) − (f3) and P verifies (P1) − (P3). For all λ sufficiently

large, (P ) has at least m− 1 non-negative weak solutions u1, · · · , um−1 ⊂ L∞(RN ) such that ak−1 < |uk|∞ ≤ ak,

for k = 2, · · · ,m.

In the end, we show that condition (f3) is a necessary condition to existence of solution to problem (P).

Theorem 2.2. Assume that f(0) > 0 and−∆u = P (x)f(u) in RN

lim|x|→∞ u(x) = 0,(P0)

has a nonnegative weak solution u such that |u|∞ ∈ (ak, ak+1], for some k ∈ 1, · · · ,m− 1, then for such k∫ ak+1

ak

f(s)ds > 0. (1)

References

[1] K.J. Brown and H. Budin, Multiple positive solutions for a class of nonlinear boundary valueproblems, J. Math.

Anal. Appl., 60, 329-338 (1977)

[2] K.J. Brown and H. Budin, On the existence of positive solutions for a class of semilinear elliptic boundary

value problems, SIAM J. Math. Anal., 60, 875-883 (1979)

[3] F.J.S.A. Correa, M.L. Carvalho, J.V.A. GonA§alves and K.O. Silva, Positive solutions of strongly nonlinear

elliptic problems, Asymptotic Analysis, 93, (2015) 1-20 DOI 10.3233/ASY-141278

[4] D.G. De Figueiredo, On the existence of multiple ordered solutions for nonlinear eigenvalue problems, Nonlinear

Anal. 11, 481-492 (1987)

[5] P. Hess, On multiple solutions of nonlinear elliptic eigenvalue problems, Comm. Partial Differential Equations,

6 (N. 8), 951-961 (1981)

[6] N. H. Loc and K. Schmitt, On Positive solutions of Quasilinear Elliptic Equations. Differential Integral

Equations, 22, Number 9/10 (2009), 829-842.


EXISTENCE AND APPROXIMATION OF SOLUTIONS FOR A CLASS OF DEGENERATE

ELLIPTIC EQUATIONS WITH NEUMANN BOUNDARY CONDITION

ALBO CARLOS CAVALHEIRO1

1Department of Mathematics, State University of Londrina, Brazil, [email protected]

Abstract

In this work we study the equation Lu = f , where L is a degenerate elliptic operator, with Neumann

boundary condition in a bounded open set Ω. We prove the existence and uniqueness of weak solutions in

the weighted Sobolev space W1,2(Ω, ω) for the Neumann problem. The main result establishes that a weak

solution of degenerate elliptic equations can be approximated by a sequence of solutions for non-degenerate

elliptic equations.

1 Introduction

In this paper, we prove the existence and uniqueness of (weak) solutions in the weighted Sobolev space W 1,2(Ω, ω)

for the Neumann problem

(P )

Lu(x) = f(x) in Ω,

⟨A(x)∇u, η(x)⟩ = 0 on ∂Ω,

where η(x) = (η1(x), ..., ηn(x)) is the outward unit normal to ∂Ω at x, ⟨., .⟩ denotes the usual inner product in Rn,

the symbol ∇ indicates the gradient and L is a degenerate elliptic operator

Lu = −n∑

i,j=1

Dj

(aij Diu

)+

n∑i=1

biDiu+ g u + θ uω, (1)

with Dj =∂

∂xj, (j = 1, ..., n), θ is positive a constant, the coefficients aij , bi and g are measurable, real-valued

functions, the coefficient matrix A(x) = (aij(x)) is symmetric and satisfies the degenerate ellipticity condition

λ|ξ|2ω(x)≤⟨A(x)ξ, ξ⟩≤Λ|ξ|2ω(x) (2)

for all ξ∈Rn and almost every x∈Ω⊂Rn a bounded open set with piecewise smooth boundary (i.e., ∂Ω∈C0,1), ω

is a weight function (that is, locally integrable and nonnegative function on Rn), λ and Λ are positive constants

2 Main Results

The main purpose of this paper (see Theorem 2.2) is to establish that a weak solution u∈W 1,2(Ω, ω) for the

Neumann problem (P ) can be approximated by a sequence of solutions of non-degenerate elliptic equations.

Theorem 2.1. Let Ω⊂Rn be a bounded open set with boundary ∂Ω∈C0,1. Suppose that

(H1) ω ∈A2;

(H2) f/ω ∈L2(Ω, ω);

(H3) bi/ω ∈L∞(Ω) (i=1,...,n) and g/ω ∈L∞(Ω).

Then, there exists a constant C > 0 such that for all θ≥C the Neumann problem (P) has a unique solution

u∈W 1,2(Ω, ω). Moreover, we have that ∥u∥W 1,2(Ω,ω)≤2

λ

∥∥∥∥ fω∥∥∥∥L2(Ω,ω)

.

13

14

Proof See [2], Theorem 1.

Lemma 2.1. Let α, β > 1 be given and let ω ∈Ap (1 < p < ∞), with Ap-constant C(ω, p) and let aij = aji be

measurable, real-valued functions satisfying λ|ξ|2ω(x)≤⟨A(x)ξ, ξ⟩≤Λ|ξ|2ω(x) (see (2)). Then there exist weights

ωαβ ≥ 0 a.e. and measurable real-valued functions aαβij such that the following conditions are met.

(i) c1(1/β)≤ωαβ(x)≤ c2 α in Ω, where c1 and c2 depend only on ω and Ω.

(ii) There exist weights ω1 and ω2 such that ω1 ≤ωαβ ≤ ω2, where ωi ∈Ap and C(ωi, p) depends only on C(ω, p)

(i = 1, 2).

(iii) ωαβ ∈Ap, with constant C(ωαβ , p) depending only on C(ω, p) uniformly on α and β.

(iv) There exists a closed set Fαβ such that ωαβ≡ω in Fαβ and ωαβ∼ ω1∼ ω2 in Fαβ with equivalence constants

depending on α and β (i.e., there are positive constants cαβ and Cαβ such that cαβ ωi ≤ωαβ ≤Cαβ ωi, i = 1, 2).

Moreover, Fαβ ⊂Fα′β′ if α≤α′, β≤β′, and the complement of⋃

α,β≥ 1

Fαβ has zero measure.

(v) ωαβ→ω a.e. in Rn as α, β→∞.

(vi) λωαβ(x) |ξ|2 ≤n∑

i,j=1

aαβij (x) ξiξj ≤Λωαβ(x) |ξ|2, ∀ ξ ∈Rn and a.e. x∈Ω, and aαβij (x) = aαβji (x).

(vii) aαβij (x) = aij(x) in Fαβ.

Proof See [1], Theorem 2.1.

Theorem 2.2. Let Ω⊂Rn be a bounded open set with boundary ∂Ω∈C0,1. Suppose (H1), (H3) and

(H2∗) f/ω ∈L2(Ω, ω)∩L2(Ω, ω3).

Then the unique solution u∈W 1,2(Ω, ω) of problem (P ) is the weak limit in W 1,2(Ω, ω1) of a sequence of solutions

um ∈W 1,2(Ω, ωm) of the problems

(Pm)

Lmum(x) = fm(x), in Ω,

⟨Am(x)∇um, η(x)⟩ = 0, on ∂Ω,

with Lmum = −n∑

i,j=1

Dj(ammij Dium) +

n∑i=1

bmiDium + gm um + θ um ωm, fm = f (ωm/ω)1/2, gm = g ωm/ω,

bmi = bi ωm/ω and ωm = ωmm (where ωmm, ammij and ω1 are as Lemma 2.1 and Am(x) =

(ammij (x)

)).

Proof See [2], Theorem 2.

References

[1] cavalheiro, a. c. - An approximation theorem for Ap-weights, MathLAB Journal, 7 (2020), 34-42.

[2] Cavalheiro, a. c. - Existence and approximation of solutions for a class of degenerate elliptic equations with

Neumann boundary condition., Note Mat. 40 (2020) no. 2, 63-81.


ON A PRECISE SCALING TO CAFFARELLI-KOHN-NIRENBERG INEQUALITY

ALDO BAZAN1 & WLADIMIR NEVES2

1Instituto de Matematica e Estatıstica, UFF, Niteroi, Brasil, [email protected],2Instituto de Matematica, UFRJ, RJ, Brasil, [email protected]

Abstract

We analyze the Caffarelli-Kohn-Nirenberg inequality in the Euclidean setting, in the non-sharp case. Due to

a new parameter introduced, this inequality presents two distinguishable ranges: in one of them, it is shown to

be the interpolation between weighted Hardy and weighted Sobolev inequalities; in the other range, the constant

is not necessarily bounded for all value of the parameters. In the former case, it is obtained a constant that

depends of the new parameter.

1 Introduction

In this work, we consider the general form of Caffarelli-Kohn-Nirenberg inequality in the non-sharp case, as appeared

in [2]: (∫Rn

∥x∥γr|u|rdx)1/r

≤ C(∫

Rn

∥x∥αp ∥∇u∥p dx)a/p(∫

Rn

∥x∥βq|u|qdx)(1−a)/q

, (1)

where the real parameters p, q, r, α, β, γ, satisfy

p, q ≥ 1, r > 0 and γr, αp, βq > −n. (2)

From a dimensional balance of (1), it follows that

1

r+γ

n= a

(1

p+α− 1

n

)+ (1 − a)

(1

q+β

n

), (3)

where a ∈ [ 0, 1] and

γ = a σ + (1 − a)β (4)

for some parameter σ. In particular, if a > 0, then σ ≤ α. Moreover, if a > 0 and also

1

p+α− 1

n=

1

r+γ

n,

then σ ≥ α − 1. These are necessary and sufficient conditions for (1), as it was proved in [2]. Further, for any

compact set in the parameter space, such that, (P), (3) and (α − 1) ≤ σ ≤ α, the positive constant C in (1) is

bounded.

Here the analyze of Caffarelli-Kohn-Nirenberg inequality relies in a suitable introduced parameter s defined by

s :=np

n− p(σ − (α− 1)), (5)

and we will be focused on the sufficiency.

15

16

2 Main Results

The main problem to make an analysis of the inequality (1) is the interpolation between the parameters on the

right side of the inequality. The following result simplifies the analysis of that interpolation.

Proposition 2.1. Assume conditions (P) and (4). If there exists a constant C > 0, such that(∫Rn

∥x∥σs |u(x)|s dx)1/s

≤ C

(∫Rn

∥x∥αp ∥∇u(x)∥p dx)1/p

, (6)

then the Caffarelli-Kohn-Nirenberg inequality (1) holds with the same constant.

Now, the following result shows the existence of the constant C for the inequality (6).

Theorem 2.1. Let p ≥ 1, α, and σ be such that αp > −n, σ ≤ α. Consider s as defined in (5) satisfying σs > −n.Then, there exists C > 0, such that (6) holds.

The proof of this theorem, as appeared in [1], shows that the value of constant C depends of the values of the

parameter s.

References

[1] bazan, a. and neves, w. - On a Precise Scaling to Caffarelli-Kohn-Nirenberg Inequality Acta Appl Math,

171 n.1, 1-15, 2021.

[2] caffarelli, l., kohn, r., nirenberg, l. - First Order interpolation inequalities with weights. Comp. Math.,

53 n.3, 259–275, 1984.

[3] folland, g. b. - Real Analysis., John Wiley & Sons, Second edition, 1999.


FINSLER DOUBLE PHASE PROBLEMS INVOLVING CRITICAL SOBOLEV NONLINEARITIES

CSABA FARKAS1, ALESSIO FISCELLA2 & PATRICK WINKERT3

1Sapientia Hungarian University of Transylvania, Hungary, [email protected],2IMECC, UNICAMP, SP, Brasil, [email protected],

3Technische Universitat Berlin, Germany, [email protected]

Abstract

In this talk, we discuss about recent results for double phase problems via variational methods. More

precisely, our problems are driven by the so-called Finsler double phase operator given by

div(F p−1(∇u)∇F (∇u) + a(x)F q−1(∇u)∇F (∇u)

)for u ∈W 1,H,F (Ω),

set on an appropriate Musielak-Orlicz-Sobolev space W 1,H,F (Ω), with F a positively homogeneous Minkowski

norm, 1 < p < q <∞ and a ∈ L∞(Ω) such that a(x) ≥ 0 a.e. in Ω. For the first time in literature, we deal with

critical Sobolev nonlinearities on a double phase setting. These nonlinear terms make the study of the energy

functional more intriguing, considering the lack of compactness of the critical Sobolev embedding forW 1,H,F (Ω).

Under suitable assumptions for weight a, exponents p and q, we are able to provide the existence of at least one

solution for our problems.

1 Introduction

In the paper [2], C. Farkas and P. Winkert studied for the first time in literature a double phase problem involving

a critical Sobolev nonlinearity. More precisely, they considered problem− div

(F p−1(∇u)∇F (∇u) + a(x)F q−1(∇u)∇F (∇u)

)= up

∗−1 + λ(uγ−1 + g(x, u)

)in Ω,

u > 0 in Ω,

u = 0 on ∂Ω,

(1)

where the main differential operator is the so-called Finsler double phase operator, with F : RN → [0,∞) a

positively homogeneous Minkowski norm. Here, they also assumed that Ω ⊂ RN is an open bounded set with

Lipschitz boundary, λ > 0 is a real parameter, p∗ = Np/(N − p), exponent γ ∈ (0, 1), g is a suitable subcritical

term, while 2 ≤ p < q < N and the following assumption holds true

(A)q

p< 1 +

1

N, while a : Ω → [0,∞) is Lipschitz continuous.

Because of the presence of an operator with non-standard growth, the natural functional space where finding

solutions of (P ) is the homogeneous Musielak-Orlicz-Sobolev space W 1,H,F0 (Ω), set with respect to F and to

function H(x, t) := tp + a(x)tq, with (x, t) ∈ Ω × [0,∞). In order to handle the critical Sobolev nonlinearity and

the non-differentiable singular term in (P ), C. Farkas and P. Winkert worked with a local analysis on a suitable

closed convex subset of W 1,H,F0 (Ω), by strongly assuming that 2 ≤ p < q < N .

Following this direction, joint with C. Farkas and P. Winkert, we were able to generalize problem (P ) considering

a nonlinear boundary condition and above all covering the complete situation 1 0 in Ω,(F p−1(∇u)∇F (∇u) + a(x)F q−1(∇u)∇F (∇u)

)· ν = up∗−1 + g2(x, u) on ∂Ω,

(2)

17

18

where Ω ⊂ RN is still an open bounded set with Lipschitz boundary, λ > 0 is a real parameter, exponent γ ∈ (0, 1),

p∗ = Np/(N − p) and p∗ = (N − 1)p/(N − p), 1 < p < q < N and we have

(A) q < p∗, while a : Ω → [0,∞) with a ∈ L∞(Ω).

In order to cover the complete situation, here F : RN → [0,∞) is a positively homogeneous Minkowski norm

satisfying

(F ) the reversibility rF = maxξ =0

F (−ξ)F (ξ)

is finite.

Since we look for positive weak solutions of (2), here g1 : Ω × R → R and g2 : ∂Ω × R → R are Caratheodory

functions verifying

(G) g1(x, t) = g2(x, t) = 0 for all t ≤ 0 and for a.e. x ∈ Ω and x ∈ ∂Ω, respectively. Furthermore, there exist

θ1 ∈ (1, p), r1 ∈ [p, p∗), r2 ∈ (p, p∗) as well as nonnegative constants a1, a2 and b1 such that

g1(x, t) ≤ a1tr1−1 + b1t

θ1−1 for a. e.x ∈ Ω and for all t ≥ 0,

g2(x, t) ≤ a2tr2−1 for a. e.x ∈ ∂Ω and for all t ≥ 0.

2 Main Results

In this talk, I will introduce the existence result for (2), proved in [1] and stated below.

Theorem 2.1. Let Ω ⊂ RN be an open bounded set with Lipschitz boundary, and let γ ∈ (0, 1). Let 1 0 such that for any λ ∈ (0, λ∗) problem

(2) admits a positive weak solution.

Inspired by [2], for the proof of myth 2.1 we used a minimization argument on a suitable closed convex subset of

Musielak-Orlicz-Sobolev space W 1,H,F (Ω). However, in order to cover the situation 1 < p < q < N , we exploited

also a truncation argument which forces the new assumption (F ).

References

[1] C. Farkas, A. Fiscella and P. Winkert - Singular Finsler double phase problems with nonlinear boundary

condition, Adv. Nonlinear Stud. (2021) DOI: https://doi.org/10.1515/ans-2021-2143

[2] C. Farkas and P. Winkert - An existence result for singular Finsler double phase problems, J. Differential

Equations 286 (2021) 455-473.


SCHRODINGER EQUATIONS WITH VANISHING POTENTIALS INVOLVING BREZIS-KAMIN

TYPE PROBLEMS

J. A. CARDOSO1, P. CERDA2, D. S. PEREIRA3 & P. UBILLA4

1Departamento de Matematica, UFS, Aracaju, SE, Brasil, [email protected],2Departamento de Matematica y C. C., USACH, Casilla 307, Correo 2, Santiago, Chile,

3Unidade Academica de Matematica, UFCG, Campina Grande, PB, Brasil,4Departamento de Matematica y C. C., USACH, Casilla 307, Correo 2, Santiago, Chile

Abstract

We prove the existence of a bounded positive solution for the following stationary Schrodinger equation

−∆u+ V (x)u = f(x, u), x ∈ Rn, n ≥ 3,

where V is a vanishing potential and f has a sublinear growth at the origin (for example if f(x, u) is a concave

function near the origen). For this purpose we use a Brezis-Kamin argument included in [3]. In addition, if f

has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce

an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For

instance, our approach can be applied for nonlinearities of the type ρ(x)f(u) where f is a concave-convex

function and ρ satisfies the (H) property introduced in [3]. We also note that we do not impose any integrability

assumptions on the function ρ, which is imposed in most works.

1 Introduction

We study existence of positive solutions for the semilinear Schrodinger equations

− ∆u+ V (x)u = f(x, u) in Rn, n ≥ 3, (P)

where V is a continuous and nonnegative vanishing potential, that is, lim|x|→∞ V (x)

= 0, and f(x, u) is a Caratheodory function. The main models of f(x, u) studied here are

I. ρ(x)uq II. λρ(x)(u+ 1)p and III. λρ(x)(uq + up),

where 0 < q < 1 < p < n+2n−2 and ρ satisfies the property (H) introduced by Brezis and Kamin [3]: a function

ρ ∈ L∞loc(Rn), ρ ≥ 0, has the property (H) if the linear problem

− ∆u = ρ in Rn (1)

has a bounded solution.

1.1 Two solutions involving nonlinearities of type II

Assuming ρ ∈ L∞(Rn), ρ ≥ 0, ρ = 0 such that

0 < ρ(x) ≤ k

1 + |x|βin Rn, (Hρ)

for constants k > 0 and β > 2 we will establish the existence of at least two solutions for two families of superlinear

Schrodinger equations. We observe that ρ is integrable only for β > n, but here we also consider 2 < β ≤ n.

19

20

The first nonlinear Schrodinger equation such that we obtain two positive solutions is the following:−∆u+ V (x)u = λρ(x)(u+ 1)p in Rn,

u > 0 in Rn,

u(x) → 0 as |x| → ∞,

(Pλ,p)

where 1 < p < 2∗ − 1 and 2∗ := 2n/(n − 2), n ≥ 3, is the critical Sobolev exponent. Our main result concerning

Problem (Pλ,p) is the following.

Theorem 1.1. Assume that ρ satisfies (Hρ) and V is a nonnegative and continuous potential such that

a

1 + |x|α≤ V (x) ≤ A

1 + |x|α, for all x ∈ Rn, (Hα

V )

for some constants a,A > 0, α ∈ (0, 2], with α + β > 4. Then, there exists Λ > 0 such that problem (Pλ,p) has at

least two positive solutions u1,λ < u2,λ in Rn, for any λ ∈ (0,Λ). Furthermore

u1,λ(x) ≤ cλ U(x) for all x ∈ Rn,

where cλ → 0 as λ→ 0.

References

[1] H. Brezis and S. Kamin - Sublinear elliptic equations in RN , Manuscripta Math., 74 (1992), 87–106.

[2] J. A. Cardoso, P. Cerda, D. Pereira, P Ubilla - Schrodinger equations with vanishing potentials

involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems, 41 (2021), 2947 –2969.


PROBLEMAS DO TIPO HENON COM O OPERADOR 1-LAPLACIANO

ANDERSON DOS S. GONZAGA1 & MARCOS T. O. PIMENTA2

1Departamento de Matematica - UNESP- Ibilce, SP, Brasil, [email protected],2Departamento de Matematica - UNESP- Fct, SP, Brasil, [email protected]

Abstract

Estudamos, neste trabalho, uma classe de problemas de Dirichlet que envolve a equacao do tipo Henon com

o operador 1− Laplaciano na bola unitaria B ⊂ RN . Para isso, provamos a imersao entre os espacos BVrad(B)

e os espacos Lr(B) com peso. Atraves de um metodo de aproximacoes do problema original por problemas

envolvendo o operador p− Laplaciano, provamos a existencia de solucoes radialmente simetricas.

1 Introducao

Na referencia [1], Henon propos o seguinte problema

− ∆u = |x|α|u|q−2u em RN, (1)

para estudar a estabilidade dos aglomerados globulares, que sao agrupamentos de estrelas aproximadamente

esfericos, em astrofısica. Desde entao, varios pesquisadores estudaram inumeros tipos de generalizacoes desta

equacao. Nosso objetivo aqui e estudar a existencia de solucao radial para o seguinte problema do tipo Henon

envolvendo o operador 1-Laplaciano: −∆1u = |x|αf(u) em B

u = 0 sobre ∂B,(2)

em que B = B(0, 1) ⊂ RN , N ≥ 2, α > 0 e f e uma funcao localmente Holder contınua onde f(s) ≥ 0 se s > 0 e

que satisfaz:

(f1) existe a > 0 tal que

lim sups→0

f(s)

|s|a= 0, uniformemente em x ∈ B,

(f2) existem C > 0 e q ∈ (0, 1∗α − 1) tais que

|f(s)| ≤ C(1 + |s|q), ∀s ∈ R,

onde 1∗α = N+αN−1 ,

(f3) existe κ > 1 tal que

0 < κF (s) ≤ f(s)s,

para todo s > 0, onde F (t) =∫ t

0f(s)ds.

Por meio de um esquema de aproximacao por solucoes de problemas envolvendo o operador p−laplaciano, mostramos

a existencia de uma solucao para o problema (1). Para isso, nos baseamos em [2] e provamos a existencia das solucoes

radiais up ∈W 1,p0,rad(B) no nıvel do Passo da Montanha do seguinte problema:−∆pu = |x|αf(u) em B

u = 0 sobre ∂B,(3)

21

22

em seguida, usamos alguns argumentos de [3] e demostramos que (up) converge para u, quando p → 1+ onde

u ∈ BVrad(B) satisfaz (1), sendo necessario nesta ultima argumentacao a utilizacao das imersoes de BVrad(B) em

Lrα(B).

2 Resultados Principais

Theorem 2.1. Seja α > 0, entao a imersao BVrad(B) → Lrα(B) e contınua para 1 ≤ r ≤ 1∗α = N+α

N−1 e compacta

para 1 ≤ r < 1∗α = N+αN−1 .

Theorem 2.2. Supondo N ≥ 2 e que f satisfaz as condicoes (f1) − (f3), entao existe uma solucao nao-negativa

u ∈ BVrad(B) de (1).

References

[1] henon, m. - Numerical experiments on the stability of spherical stellar systems. Astronomy and Astrophysics,

24, 229 - 238, 1973.

[2] ni, w. m. - A nonlinear Dirichlet problem on the unit ball and its applications. Indiana University Mathematics

Journal, 31, No. 6, 801-807, 1982.

[3] segura, s. and molino, a. - Elliptic equations involving the 1−Laplacian and a subcritical source term.

Nonlinear Analysis, 168, 50 - 66, 2018.


ON A CLASS OF ELLIPTIC SYSTEMS OF THE HARDY-KIRCHHOFF TYPE IN RN

AUGUSTO C. R. COSTA1, OLIMPIO H. MIYAGAKI2 & FABIO R. PEREIRA3

1Instituto de Ciencias Exatas e Naturais, Faculdade de Matematica, UFPA, PA, Brasil, [email protected],2Centro de Ciencias Exatas e de Tecnologia, Departamento de Matematica, UFSCar, SP, Brasil, [email protected],

3Instituto de Ciencias Exatas, Departamento de Matematica, UFJF, MG, Brasil, [email protected]

Abstract

In this work we consider a class of critical variational systems in RN of the Hardy-Kirchhoff type involving

the fractional Laplacian operator. By imposing some conditions on the nonlinearity as well as in the potencial,

we recover the compactness combining arguments used in Alves and Souto [1] and in Brezis and Nirenberg [2].

Only monotonicity conditions are employed, without imposing any coercivity condition on the potencial, which

can tend to zero at infinity. Our result is closely related to that obtained by Fiscella, Pucci and Zhang [3].

1 Introduction

In this work we study a class of critical systems in RN of the Hardy-Kirchhoff type involving the fractional Laplacian

operator of the formM(∥u∥2)( LV u) =

λp

p+ qK(x)|u|p−2u|v|q +

α

2∗s|u|α−2u|v|β in RN ,

M(∥v∥2)( LV v) =λq

p+ qK(x)|u|p|v|q−2v +

β

2∗s|u|α|v|β−2v in RN ,

(1)

where LV w ≡ (−∆)sw− σ w|x|2s + V (x)w, with σ > 0 (to be chosen), λ > 0 and 0 < s < 1, N > 2s. We assume that

p, q, α, β > 1 are such that 4 < p+ q < α+ β = 2∗s = 2NN−2s and suppose that the Kirchhoff function M : R+ → R+

is given by M(t) = a + bt, a, b > 0, K and V are positive continuous functions and (−∆)s fractional Laplacian,

which is defined, up to a normalization constant, as

(−∆)sϕ(x) = 2 limϵ→0+

∫RN\Bϵ(x)

ϕ(x) − ϕ(y)

|x− y|N+2sdy, x ∈ RN , ϕ ∈ C∞

0 (RN ),

and

∥w∥2 = CN,s

∫ ∫R2N

|w(x) − w(y)|2

|x− y|N+2sdxdy − σ

∫RN

|w|2

|x|2sdx+

∫RN

V (x)w2dx.

Assumptions on V and K:

(i) (sign of V and K) V,K are continuous, V,K > 0 on RN and K ∈ L∞(RN );

(ii) (decay of K) If An is a sequence of Borel sets of RN with |An| ≤ R for some R > 0,

limr→∞

∫An∩Bc

r(0)

K(x)dx = 0, uniformly with respect to n ∈ N. (2)

The above type of (V,K) condition, it was introduced by Alves-Souto [1].

2 Main Result

Theorem 2.1. In addition to (V,K), suppose 4 0 the problem (1) possesses a positive solution.

23

24

References

[1] alves, c. o. and souto, m. a. s. - Existence of solutions for a class of nonlinear Schrodinger equations with

potential vanishing at infinity, J. Differential Equations 254 (2013), 1977–1991.

[2] brezis, h, and nirenberg, l. - Positive solutions of nonlinear elliptic equations involving critical Sobolev

exponents. Comm. Pure Appl. Math. 36 (1983), no. 4, 437-477.

[3] fiscella, a., pucci, p. and zhang, b. - p−fractional Hardy–Schrodinger–Kirchhoff systems with critical

nonlinearities, Adv. Nonlinear Anal. 8 (2019), 1111–1131.


A HESSIAN-DEPENDENT FUNCTIONAL WITH FREE BOUNDARIES AND APPLICATIONS TO

MEAN-FIELD GAMES

JULIO C. CORREA1 & EDGARD A. PIMENTEL2

1Department of Mathematics, Catholic University of Rio de Janeiro, Rio de Janeiro-RJ, Brazil, [email protected],2Department of Mathematics, Catholic University of Rio de Janeiro, Rio de Janeiro-RJ, Brazil, [email protected]

Abstract

We study a Hessian-dependent functional driven by a fully nonlinear operator, which associated Euler-

Lagrange equation is a fully nonlinear mean-field game with free boundaries. Our findings include the existence of

solutions to the mean-field game, together with Holder continuity of the value function and improved integrability

of the density. In addition, we derive a free boundary condition and prove that the reduced free boundary is a

set of finite perimeter.

1 Introduction

We examine Hessian-dependent functionals of the form

FΛ,p[u] :=

∫B1

F (D2u)pdx+ Λ|u > 0 ∩B1|, (1)

where F : S(d) → R is a uniformly elliptic operator, Λ > 0 is a fixed constant, and p > d/2. The functional in

(1) is inspired by the usual one-phase Bernoulli problem, driven by the Dirichlet energy. To a limited extent, we

understand FΛ,p as a Hessian-dependent counterpart of that problem. See [1]; see also [2].

The analysis of (1) relates closely with the systemF (D2u) = m1

p−1 in B1 ∩ u > 0(Fi,j(D

2u)m)xixj

= 0 in B1 ∩ u > 0,(2)

where Fi,j(M) denotes the derivative of F with respect to the entry mi,j of M . Here, the unknown is a pair (u,m)

solving the problem in a sense we make precise further. In fact, the system in (2) amounts to the Euler-Lagrange

equation associated with (1). Furthermore we notice that (2) satisfies an adjoint structure. Due to such a distinctive

pattern, we refer to (2) as a fully nonlinear mean-field game with free boundary.

The interesting aspect in (2) concerns the appearance of a free boundary. At least heuristically, the game is

played only in the regions where the value function is strictly positive. Combined with the free boundary condition,

(2) models a game in which players optimize in the region where the value function is positive and might face

extinction according to a flux condition endogenously determined.

2 Main Results

Since one can state the Euler-Lagrange equation associated with (1) in terms of a fully nonlinear mean-field game

system with free boundaries, our analysis of the existence of solutions to (2) relies on the existence of minimizers of

(1) and their interplay with the notion of a solution of a fully nonlinear mean-field game. In the sequel we define a

solution of the mean-filed game (2).

Definition 2.1 (Solution for the MFG system). The pair (u,m) is a weak solution to (2) if the following hold:

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26

1. We have u ∈ C(B1) ∩W 1,pg and m ∈ L1(B1), with m ≥ 0;

2. The function u is an Lp-viscosity solution to

F (D2u) = m1

p−1 in B1 ∩ u > 0;

3. The function m is a weak solution to(Fij(D

2u)m)xixj

= 0 in B1 ∩ u > 0.

The definition of Lp-viscosity solution is necessary since Lp-functions might not be defined at the points where

the usual conditions must be tested. For a comprehensive account of this notion, we refer the reader to [6].

The first contribution in our recent preprint [7] is to prove the existence of solutions for the mean-field game system

(2). We report our findings in the following

Theorem 2.1 (Existence and regularity of solutions). Suppose F is a convex, uniformly elliptic operator, satisfying

a suitable growth condition, and g ∈W 2,p non-negative. Then there exists a solution (u,m) to (2). In addition, fix

α ∈ (0, 1). We have u ∈ Cαloc(B1) and there exists C > 0 such that

∥u∥Cα(B1/2) ≤ C∥g∥W 2,p(B1).

The constant C > 0 depends on the exponent α.

Once we have established the existence of solutions for (2) and produced a regularity result, we examine the

free boundary. We resort to a variation of the functional and derive a free boundary condition. We summarize our

findings in this direction in the following result.

Theorem 2.2 (Free boundary condition and finite perimeter). Let u ∈ W 2,ploc (B1) ∩W 1,p

g (B1) be a minimizer for

(1), for p > d/2. Suppose F is a convex, uniformly elliptic operator, satisfying a suitable growth condition, and

g ∈W 2,p non-negative. Then ∂∗u > 0 is a set of finite perimeter. Suppose in addition u ∈ C2(B1); then∫∂u>0

(F (D2u)p−1Fij(D

2u)xiuxj

− Λ

2p

)⟨ξ, ν⟩ dHd−1 = 0 (1)

for every ξ ∈ C∞c (B1,Rd).

References

[1] alt. w. and caffarelli, l. - Existence and regularity for a minimum problem with free boundary, J. Reine

Angew. Math. 325 (1981), 105-144.

[2] caffarelli, l. and salsa, s. - A geometric approach to free boundary problems, volume 68 of Graduate

Studies in Mathematics. American Mathematical Society, Providence, RI, 2005.

[3] lions, p. l. - Cours au coll’ege de france. www.college-de-france.fr.

[4] andrade, p. and pimental, e. - Stationary fully nonlinear mean-field games.

[5] chowdhury, i. jakobsen, e. r. and krupski, m. - On fully nonlinear parabolic mean field games with

examples of nonlocal and local diffusions, 2021.

[6] caffarelli, l. crandall, c.g. kocan, m. and swiech, a. - On viscosity soluntions of fully nonlinear

equations with measureble ingredients. Comm. Pure Appl. Math., 49(4): 365-397, 1996.

[7] correa, j. c. and pimental e. - A Hessian-dependent functional with free boundaries and apllpications to

mean-field games. arxiv preprint arKiv:2107.00743.


POSITIVE SOLUTIONS FOR A CLASS OF FRACTIONAL CHOQUARD EQUATION IN

EXTERIOR DOMAIN

CESAR T. LEDESMA1 & OLIMPIO H. MIYAGAKI2

1Departamento de Matematicas, Universidad Nacional de Trujillo, Trujillo, Peru, ctl [email protected], C. T. L. received

research grants from CONCYTEC, Peru, 379-2019-FONDECYT “ASPECTOS CUALITATIVOS DE ECUACIONES

NO-LOCALES Y APLICACIONES,2 Departamento de Matematica, Universidade Federal de Sao Carlos, 13565-905 - Sao Carlos - SP - Brazil,

[email protected], O. H. M. received research grants from CNPq/Brazil Proc 307061/2018-3 and FAPESP Proc

2019/24901-3.

Abstract

This work concerns with the existence of positive solutions for the following class of fractional elliptic

problems, (−∆)su+ u =

(∫Ω

|u(y)|p

|x− y|N−αdy

)|u|p−2u, in Ω

u = 0, RN \ Ω(1)

where s ∈ (0, 1), N > 2s, α ∈ (0, N), Ω ⊂ RN is an exterior domain with smooth boundary ∂Ω = ∅ and

p ∈ (2, 2∗s). The main feature from problem (1) is the lack of compactness due to the unboundedness of the

domain and the lack of the uniqueness of solution of the limit problem. To overcome the loss these difficulties

we use splitting lemma combined with careful investigation of limit profiles of ground states of limit problem.

In recent years great attention has been devoted to the study of elliptic equations involving the fractional

Laplacian operator. It appears in many models arising from concrete applications in Biology, Physics, Game

Theory and Financial Mathematics, see [5, 8].

Recently, fractional elliptic equations like

(−∆)su+ ωu = (Kα ∗ |u|p)|u|p−2u, u ∈ Hs(RN ), (2)

where ω > 0, α ∈ (0, N), p > 1, s ∈ (0, 1) and Kα(x) = |x|α−N was considered. When s = 1/2, Frank and Lenzmann

[7] have used problem (1) to model the dynamics of pseudo-relativistic boson stars. Indeed they considered the

existence of ground state solution of the following equation:√−∆u + u = (K2 ∗ |u|2)u, u ∈ H1/2(R3), u > 0.

Moreover, in [6] the author showed that the dynamical evolution of boson stars is described by the nonlinear

evolution equation i∂tψ =√

−∆ +m2ψ − (K2 ∗ |ψ|2)ψ (m ≥ 0) for a field ψ : [0, T ) × R3 → C. In [2], d’Avenia

et. all. considered problem (1) and obtained regularity, existence, nonexistence, symmetry and decay properties of

the corresponding solutions.

When s = 1, Moroz and Schaftingen [9] considered the following equation in exterior domains

− ∆u+W (x)u = (Kα ∗ |u|p)|u|p−2u, u ∈ H10 (Ω). (3)

They showed that problem (2) does not have nontrivial nonnegative super solutions. Moreover Clapp and Salazar [4],

under symmetry conditions on unbounded exterior domain Ω and W established the existence of a positive solution

and multiple sign changing solutions for (2). When Ω has no symmetry, the study becomes more complicated, see

[3]. After a bibliographic review, we have observed, up to our knowledge, that there is no results in the literature,

for a version of problem (2), also for the fractional case, without any symmetry conditions.

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28

In the fractional case, recently Alves et. al [1] studied the problem

(−∆)su+ u = |u|p−2u, in Ω, u ≥ 0, in Ω and u ≡ 0, u = 0,RN \ Ω (4)

where p ∈ (2, 2∗s) and Ω is an exterior domain with (non-empty) smooth boundary ∂Ω. They proved that (3)

does not have a ground state solution, which becomes a difficulty in dealing with the problem. As in [3], the

authors analyzed the behavior of Palais-Smale sequences, obtaining a precise estimate of the energy levels where

the Palais-Smale condition fails, which made possible to show that without any symmetry assumption the problem

(3) has at least one positive solution, for RN \ Ω small enough. We note that, a key point to prove the results of

existence is the uniqueness up to a translation of positive solution of the equation at infinity associated with (3)

given by (−∆)su + u = |u|p−2u, in RN . We recall that we did not find in the literature any paper dealing with

the existence of non negative solutions for Problem (P ) in exterior domains. The main feature from problem (P )

is the lack of compactness due to the unboundedness of the domain and the lack of the uniqueness of solution of

the limit problem

(−∆)su+ u =

(∫RN

|u(y)|p

|x− y|N−αdy

)|u|p−2u, x ∈ RN . (5)

To overcome the loss of uniqueness we investigate limit profiles of ground states of (5) as α→ 0. This leads to the

uniqueness of ground states when α is closed to 0.

Our main result is the following.

Theorem 0.3. There is α0 > 0 small enough and ρ > 0 such that if RN \ Ω ⊂ B(0, ρ), problem (1) has at least

one positive solution for all α ∈ (0, α0).

References

[1] C. Alves, G. Molica Bisci and C. Torres, Existence of solutions for a class of fractional elliptic problems

on exterior domains, J. Differential Equations 268, 7183-7219 (2020).

[2] P. d’Avenia, G. Siciliano and M. Squassina, On fractional Choquard equations, Mathematical Models

and Methods in Applied Sciences, 25, 8 (2015) 1447-1476

[3] V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch.

Rational Mech. Anal. 99, 283-300 (1987).

[4] M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math.

Anal. Appl. 407, 1-15 (2013)

[5] E. DiNezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull.

Sci. math. 136, 521-573 (2012).

[6] A. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math. 60, 500-545

(2007).

[7] R. Frank and E. Lenzmann, On ground states for the L2-critical boson star equation, arXiv:0910.2721.

[8] G. Molica Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems,

University Printing House, Cambridge CB2 8BS, United Kingdom 2016.

[9] V. Moroz and J. Van Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard

equations in exterior domains, J. Differ. Equ. 254, 3089-3145 (2013).

773–813 (2017).


CHOQUARD EQUATIONS VIA NONLINEAR RAYLEIGH QUOTIENT FOR CONCAVE-CONVEX

NONLINEARITIES

CLAUDINEY GOULART1, MARCOS L. M. CARVALHO2 & EDCARLOS D. SILVA3

1Universidade Federal de Jataı, GO, Brasil, [email protected],2Instituto de Matematica, UFG, GO, Brasil, marcos leandro [email protected],

3Instituto de Matematica, UFG, GO, Brasil, [email protected]

Abstract

It is established existence of ground and bound state solutions for Choquard equation considering concave-

convex nonlinearities in the following form−∆u+ V (x)u = (Iα ∗ |u|p)|u|p−2u+ λ|u|q−2u in RN ,

u ∈ H1(RN )

where λ > 0, N ≥ 3, α ∈ (0, N). The potential V is a continuous function and Iα denotes the standard Riesz

potential. Assume also that 1 < q < 2, 2α 0 taking into account the nonlinear

Rayleigh quotient. More precisely, there exists λ∗ > 0 such that our main problem admits at least two positive

solutions for each λ ∈ (0, λ∗]. In order to do that we combine Nehari method with a fine analysis on the nonlinear

Rayleigh quotient. The parameter λ∗ > 0 is optimal in some sense which allow us to apply the Nehari method.

1 Introduction

It is well known that existence, nonexistence and multiplicity of solutions for nonlocal elliptic problems are related

with the behavior of the nonlinearity at the origin and at infinity. In this work we shall consider semilinear elliptic

problems driven by the Choquard equation described in the following form:−∆u+ V (x)u = (Iα ∗ |u|p)|u|p−2u+ λ|u|q−2u in RN ,

u ∈ H1(RN )(1)

where λ > 0, N ≥ 3, α ∈ (0, N). The potential V is a continuous function and Iα denotes the standard Riesz

potential. Assume also that 1 < q < 2, 2α < p < 2∗α where 2α = (N + α)/N , 2∗α = (N + α)/(N − 2). Later on, we

shall consider hypotheses on V and λ. Recall that the Riesz potential can be described in the following form

Iα(x) =Aα(N)

|x|N−α, x ∈ RN and Aα(N) =

Γ(N−α2 )

Γ(α2 )π

N2 2s

,

where Γ denotes the Gamma function, see [6]. The Choquard equation has many physical applications. For example

assuming that N = 3, α = 2, p = 2, λ = 0 and V ≡ 0, Problem (1) was investigated in [8] considering the quantum

theory of a polaron at rest. It was pointed in [5] that Choquard problem is also applied in the Hartree-Fock theory

of one component plasma. It also arises in multiple particles systems [2] and quantum mechanics [7].

It is important to emphasize that nonlocal elliptic problems involving Choquard equations have been studied in

the last years taking into account several different assumptions on the potential V .

Nonlinear Rayleigh quotient have been studied in the last years, see [3, 2]. The main feature in these works is

to guarantee that there exists an extreme value λ∗ > 0 in such way that the Nehari method can be applied for each

λ ∈ (0, λ∗).

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30

2 Main Results

We are concerned with existence of ground and bound states for Problem (1) involving concave-convex nonlinearities.

In this case, we need to control the parameter λ > 0 getting our main results. In order to overcome this difficulty,

we shall consider the nonlinear Rayleigh quotient showing that there exists λ∗ > 0 such that the Nehari method

can be applied for each λ ∈ (0, λ∗]. Throughout this work we assume the following assumptions:

(Q) It holds 1 < q < 2 and p ∈ (2α, 2∗α) with 2α = (N + α)/N , 2∗α = (N + α)/(N − 2);

(V1) The function V : RN → R is continuous and there exists a constant V0 > 0 such that V (x) ≥ V0 for all x ∈ RN ;

(V2) It holds V −1 ∈ L1(RN), i.e., the function V satisfies the following integrability condition∫RN V

−1(x)dx < +∞.

Now we consider the working Banach space for our problem defined by X =v ∈ H1(RN ) :

∫RN V (x)v2dx < +∞

.

It is worthwhile to mention that the energy functional associated to Problem (1) is given by

Eλ(u) =1

2||u||2 − 1

2p

∫RN

(Iα ∗ |u|p) |u|pdx− λ

q

∫RN

|u|qdx, u ∈ X.

Using the embedding X → Lr(RN ) for each r ∈ [1, 2∗] it is well known that Eλ ∈ C1(X,R). Namely, we can

use the all machinery of variational methods in order to ensure existence and multiplicity of solutions.

In this way, we can state our main result in the following way:

Theorem 2.1. Suppose (Q) and (V1) − (V2). Then, there are 0 < λ∗ < λ∗ < ∞ such that for each λ ∈ (0, λ∗)

the Problem (1) admits at least two distinct positive solutions uλ, vλ ∈ X satisfying the following statements:

E′′λ(uλ)(uλ, uλ) > 0, E′′

λ(vλ)(vλ, vλ) < 0, Eλ(uλ) < 0. Furthermore, uλ is a ground state solution and vλ satisfies

the following statements:

(i) For each λ ∈ (0, λ∗) we obtain that Eλ(vλ) > 0;

(ii) For each λ = λ∗ we deduce that Eλ(vλ) = 0;

(iii) For each λ ∈ (λ∗, λ∗) we obtain also that Eλ(vλ) < 0.

For more details about our main results, see [1].

References

[1] M. L. M. Carvalho, Edcarlos D. Silva, C. Goulart. Choquard equations via nonlinear rayleigh quotient for

concave-convex nonlinearities, Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021113

[2] E. P. Gross, Physics of Many-Particle Systems, Vol. 1, Gordon Breach, New York, 1996.

[3] Y. Il’yasov, K. Silva, On branches of positive solutions for p-Laplacian problems at the extreme value of Nehari

manifold method, Proc. Amer. Math. Soc. 146 (2018), no. 7, 2925-2935.

[4] Y. Il’yasov, On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient, Topol. Methods

Nonlinear Anal. 49 (2017), no. 2, 683-714.

[5] E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl.

Math. 57 (2) (1977) 93-105.

[6] V. Moroz, J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl. 19 (2017), no.

1, 773-813.

[7] R. Penrose, On gravity’s role in quantum state reduction, Gen. Relativity Gravitation 28 (1996) 581-600.

[8] S. Pekar, Untersuchung Ber Die Elektronentheorie Der Kristalle, Akademie Verlag, Berlin, 1954.


SOBRE A CAMADA DE TRANSICAO INTERNA DE PROBLEMAS SEMILINEARES

NAO-HOMOGENEOS: A LOCALIZACAO DA INTERFACE

SONEGO, MAICON1

1Instituto de Matematica e Computacao, UNIFEI, MG, Brasil, [email protected]

Abstract

O objetivo deste trabalho e estudar a localizacao de camadas de tansicao interna para determinadas solucoes

de uma classe de problemas elıpticos nao-homogeneos, postos num intervalo da reta, e condicoes de fronteira de

Neumann. Nos generalizamos alguns resultados conhecidos usando tecnicas variacionais inspiradas na teoria de

Γ-convergencia. Como aplicacao, apresentamos a localizacao das camadas de transicao interna para problemas

postos em algumas variedades Riemannianas simetricas.

1 Introducao

Quando uma equacao diferencial contem um parametro pequeno multiplicando o termo com derivadas espaciais

e este este parametro vai a zero, grosseiramente falando, dizemos que a famılia de solucoes a este parametro

desenvolve uma camada de transicao interna se ela induz uma particao no domınio em duas regioes onde, exceto

por uma regiao “tubular” – a chamada interface da camada de transicao – as solucoes se aproximam de duas

funcoes pre-determinadas (uma em cada regiao). Solucoes desenvolvendo camadas de transicao interna possuem

um importante papel em muitas areas da ciencia aplicada, por exemplo: teoria da combustao, transicao de fases,

formacao de padroes, dinamica populacional, reacoes quımicas, etc.

Neste trabalho contribuımos na tarefa de fornecer a localizacao exata da interface de determinada classe de

solucoes do seguinte problema singularmente perturbadoϵ2(k(x)u′(x))′ + f(u, x) = 0, x ∈ (0, 1),

u′(0) = u′(1) = 0,(1)

onde k(·) ∈ C1(0, 1) e positivo; ϵ > 0 e um parametro positivo e f : R× [0, 1] → R e de classe C1. Assumimos que

f(·, x) tem dois zeros b1(x), b2(x) tais que b1, b2 ∈ C1(0, 1) e b1(x) < b2(x) para todo x ∈ [0, 1];

∂1f(b1(x), x) < 0 e ∂1f(b2(x), x) < 0 para todo x ∈ [0, 1];

se

F (u, x) = −∫ u

b1(x)

f(s, x)ds (2)

entao F (·, x) ≥ 0 para todo x ∈ [0, 1] e√k(·)F (·, ·) e Lipschitz contınua.

Um tıpico exemplo de uma funcao f satisfazendo as condicoes acima e

f(u, x) = −(u− b1(x))(u− a(x))(u− b2(x)), (3)

com b1(·), a(·), b2(·) ∈ C1(0, 1) e b1(x) < a(x) < b2(x) (com a ≥ (b1 + b2)/2) para todo x ∈ [0, 1]. Esta funcao

esta relacionada ao problema de Allen-Cahn nao-homogeneo que tem sua origem na teoria de transicao de fases e

e usado como modelo para diversos processos de reacao e difusao nao-lineares.

31

32

2 Resultado Principal

As solucoes de (P ) sao pontos crıticos do funcional de energia Jϵ : H1(0, 1) → R definido por

Jϵ(u) =

∫ 1

0

ϵ

2k(x)|u′|2 +

1

ϵF (u, x)dx,

onde F foi definido em (2). No entanto, nosso principal resultado requer extender este funcional para L1(0, 1); i.e.

consideramos Jϵ : L1(0, 1) → R ∪ ∞ definido por penalizacao em L1(0, 1) por

Jϵ(u) =

Jϵ(u), u ∈ H1(0, 1),

∞, u ∈ L1(0, 1)\H1(0, 1).(1)

Definicao 2.1. Uma famılia uϵ de solucoes de (P ) em C2(0, 1) ∩ C1[0, 1] e dita desenvolver uma camada de

transicao interna, quando ϵ→ 0, com interface em x ∈ (0, 1) se

uϵϵ→0→ u0 := b2χ[0,x) + b1χ[x,1] em L1(0, 1). (2)

A fim de afirmar nosso resultado principal, definimos a seguinte funcao Λ : (0, 1) → R,

Λ(x) :=

∫ b2(x)

b1(x)

√k(x)F (s, x)ds (3)

e o conjunto

Q =

x ∈ (0, 1);

∫ b2(x)

b1(x)

f(s, x)ds = 0

. (4)

O resultado principal e afirmado abaixo.

Teorema 2.1. Suponha que uma famılia uϵ de solucoes de (P ) desenvolve uma camada de transicao interna

com interface em x ∈ C, onde C ⊂ Q e a componente conexa de Q na qual x esta. Entao,

se uϵ e uma famılia de mınimos locais em L1 de Jϵ, x e um ponto de mınimo local de Λ(x) em C;

se uϵ e uma famılia de mınimos globais de Jϵ, Λ(x) = minΛ(x); x ∈ C.

Este conteudo esta presente no trabalho [1].

References

[1] sonego, m. - On the internal transition layer to some inhomogeneous semilinear problems: interface location,

Journal of Mathematical Analysis and Applications, 502, 2, 125266, 2021.


ELLIPTIC SYSTEMS INVOLVING SCHRODINGER OPERATORS WITH VANISHING

POTENTIALS.

DENILSON DA S. PEREIRA1

1Unidade Academica de Matematica, UFCG,PB, Brasil, [email protected]

Abstract

We prove the existence of a bounded positive solution of the following elliptic system involving Schrodinger

operators −∆u+ V1(x)u = λρ1(x)(u+ 1)r(v + 1)p in RN

−∆v + V2(x)v = µρ2(x)(u+ 1)q(v + 1)s in RN ,

u(x), v(x) → 0 as |x| → ∞

where p, q, r, s ≥ 0, Vi is a nonnegative vanishing potential, and ρi has the property (H) introduced by Brezis and

Kamin [3]. As in that celebrated work we will prove that for every R > 0 there is a solution (uR, vR) defined on

the ball of radius R centered at the origin. Then, we will show that this sequence of solutions tends to a bounded

solution of the previous system when R tends to infinity. Furthermore, by imposing some restrictions on the

powers p, q, r, s without additional hypotheses on the weights ρi, we obtain a second solution using variational

methods for a gradient system.

1 Introducao

We first study the existence of a bounded positive solution of the system

−∆u+ V1(x)u = λρ1(x)(u+ 1)r(v + 1)p in RN

−∆v + V2(x)v = µρ2(x)(u+ 1)q(v + 1)s in RN ,

u(x), v(x) → 0 as |x| → ∞

(Sλ,µ)

where λ, µ > 0 and p, q, r, s ≥ 0, and where Vi is a vanishing potential satisfying

ai1 + |x|α

≤ Vi(x) ≤ Ai

1 + |x|αfor all x ∈ RN , (Hα

V )

for some constants α > 0 and Ai, ai ≥ 0, i = 1, 2. The weight ρi belongs to L∞(RN ) and satisfies

0 < ρi(x) ≤ ki1 + |x|β

in RN , (Hρ)

for some constants β > 2 and ki > 0, i = 1, 2. In this work, assuming the conditions (HαV ), (Hρ) and using the

upper and lower solutions technique, we first prove the existence of a bounded positive solution of System (Sλ,µ).

As far as we know, the first work for elliptic systems using the ideas of [3], was done by Montenegro [3], where

uniqueness of solution in balls also plays an important role. Since System (Sλ,µ) in bounded domains does not have

this property, we will have to use an alternative argument that involves minimal solutions.

33

34


Let us state our first result.

Theorem 2.1. Assume that p, q, r, s ≥ 0 and in addition suppose hypotheses (Hρ) and (HαV ) hold with α ∈ (0, 2]

and α + β > 4. Then, there exists Λ > 0 such that System (Sλ,µ) has at least one bounded positive solution for

every 0 < λ, µ < Λ.

When r, s > 1 we can construct a function that is the border between the region of existence and nonexistence.

Theorem 2.2. Suppose hypotheses (Hρ) and (HαV ) hold with α ∈ (0, 2] and α + β > 4. Assume also that r, s > 1

and p, q ≥ 0. Then, there is a positive constant λ∗ and a nonincreasing continuous function Γ : (0, λ∗) → [0,∞)

such that if λ ∈ (0, λ∗) then System (Sλ,µ):

i) has at least one bounded positive solution if 0 < µ < Γ(λ) ;

ii) has no bounded positive solution if

µ > Γ(λ).

On the other hand, the second positive solution will be obtained employing variational methods. Here we will

consider the following gradient system−∆u+ V (x)u = λρ1(x)(u+ 1)r(v + 1)s+1 in RN

−∆v + V (x)v = λρ2(x)(u+ 1)r+1(v + 1)s in RN ,

u(x), v(x) → 0 as |x| → ∞,

(1)

with r, s > 1, r + s < 2∗ − 2, ρ1(x) = (r + 1)ρ(x) and ρ2(x) = (s+ 1)ρ(x).

Theorem 2.3. Suppose hypotheses (Hρ) and (HαV ) hold with α ∈ (0, 2] and α+ β > 4,

i) If r, s ≥ 0, then there exists λ∗ > 0 such that the gradient System (1) possesses at least one bounded positive

solution (u1,λ, v1,λ) for all 0 < λ < λ∗ while for r, s > 1 and λ > λ∗ there are no bounded positive solutions.

ii) If r, s > 1 and r + s < 2∗ − 2, then there exists 0 < λ∗∗ ≤ λ∗ such that the gradient System (1) possesses a

second positive solution of the form (u1,λ + u, v1,λ + v) for all 0 < λ < λ∗∗, where u, v ∈ H1(RN ).

We would like to point out that in Theorem 2.3, to show existence of a second solution we will use an auxiliary

problem which allow us to avoid imposing additional hypotheses of integrabilities on the weights ρi. We also prove

a similar result for a class of Hamiltonian system.

This is a joint work with Juan Arratia (Universidad de Santiago de Chile) and Pedro Ubilla (Universidad de

Santiago de Chile) to apper at Discrete and Continuous Dynamical Systems.

References

[1] H. Brezis and S. Kamin. Sublinear elliptic equations in RN . Manuscripta Math. 74 (1992), 87-106.

[2] J. A. Cardoso, P. Cerda, D. S. Pereira and P. Ubilla. Schrodinger Equation with vanishing potentials involving

Brezis-Kamin type problems. Discrete Contin. Dyn. Syst, 2021, 41(6): 2947-2969.

[3] M. Montenegro. The construction of principal spectral curves for Lane-Emden systems and applications. Ann.

Sc. Norm. Super. Pisa Cl. Sci. 4 serie, tome 29 (2000), 193-229.


GROUND STATES FOR FRACTIONAL LINEAR COUPLED SYSTEMS VIA PROFILE

DECOMPOSITION

J. C. DE ALBUQUERQUE1, DIEGO FERRAZ2 & EDCARLOS D. SILVA3

1Departamento de MatemA¡tica, UFPE, PE, Brasil, [email protected],2Departamento de MatemA¡tica, UFRN, RN, Brasil, [email protected],3Departamento de MatemA¡tica, UFG, GO, Brasil, [email protected]

Abstract

In this work we study existence of weak and ground state (least energy) solutions for a class of nonlocal

linearly coupled elliptic systems. We deal with nonautonomous nonlinearities that may not satisfy any kind

of monotonicity, also the related potentials may not have any kind of smoothness. In order to obtain ground

states, instead of applying the well known methods of Nehari-Pohozaev manifold, we introduce new arguments

and techniques whose are based on a Pohozaev type identity, a concentration–compactness principle and a profile

decomposition type result.

1 Introduction

In this work we study the following class of linearly coupled fractional systems(−∆)su+ V1(x)u = f1(x, u) + λ(x)v, x ∈ RN ,

(−∆)sv + V2(x)v = f2(x, v) + λ(x)u, x ∈ RN ,(S)

where (−∆)s denotes the fractional Laplacian operator for s ∈ (0, 1). Coupled elliptic systems arise in various

branches of mathematical physics and nonlinear optics.

Our main motivation to study (S) is based on the following question: is it possible to develop a general argument

to obtain ground states for the class of coupled systems (S) (in particular the scalar equation λ(x) ≡ 0), when the

involved nonlinear terms does not satisfy any conditions such as

(N)f(t)

|t|is nondecreasing on t ∈ R\0,

or the related potentials are not necessarily smooth? See [1, 3, 2] for further discussion.

Our purpose here in this work is to study System (S) inspired by the above question considering that the

nonlinearities are superlinear. Roughly speaking, we replace the use of Nehari-Pohozaev manifold method by the

use a technique based on concentration-compactness via profile decomposition for weak convergence in fractional

Sobolev spaces and the use of a Pohozaev type identity. In order to approach in this way, we assume the existence

of a limit system associated with (S) as |x| → ∞. More precisely, we first study the following nonlocal elliptic

problem (−∆)su+ V1(∞)u = f1(∞, u) + λ(∞)v, x ∈ RN ,

(−∆)sv + V2(∞)v = f2(∞, v) + λ(∞)u, x ∈ RN ,(S∞)

which is obtained by taking |x| → ∞ in (S) and comparing its minimax level with the one of (S∞). Here V1(∞),

V2(∞) and λ(∞) are constants with f1(∞, u) and f2(∞, u) being autonomous functions.

Next, for each i = 1, 2 we assume 0 < si < min1, N/2 and the following general hypotheses:

35

36

(A1) Vi(x) ≥ 0 almost everywhere (a.e.) in RN , Vi ∈ Lσi

loc(RN ), σi > N/2si and

infu∈C∞

0 (RN ):∥u∥2=1

[∫RN

|(−∆)si/2u|2dx+

∫RN

Vi(x)u2dx

]> 0.

(A2) λ ∈ L∞(RN ) and there exists δ ∈ (0, 1) such that (s.t.) |λ(x)| ≤ δ√V1(x)V2(x) a.e. x ∈ RN .

(A3) Vi(∞) := lim|x|→∞ V (x) > 0 and λ(∞) := lim|x|→∞ λ(x).

For a.e. x ∈ RN we suppose that t 7→ fi(x, t) is C1 and satisfies the following assumptions:

(H1) limt→+∞ fi(x, t)t−1 = +∞, uniformly a.e. x ∈ RN .

(H2) For every compact L ⊂ R, there exists CL > 0 s.t. |fi(x, t)| ≤ CL a.e. x ∈ RN and t ∈ L.

(H3) Let Fi(x, t) = (fi(x, t)t)/2 − Fi(x, t), then

infx∈RN

[inf

a≤|t|≤bFi(x, t)

]> 0, ∀ b > a > 0.

(H4) There exists qi > N/(2si), ai > 0 and Ri > 0 such that

|f(x, t)|qi ≤ aiFi(x, t)|t|qi , ∀ |t| > Ri.

(H5) For any given ε > 0, there exist Cεi > 0 and pεi ∈ (2, 2∗si) such that∣∣∣∣∂fi∂t (x, t)

∣∣∣∣ ≤ ε+ Cεi |t|pεi−2, for a.e. x ∈ RN and ∀ t ∈ R.

(H6) fi(∞, t) := lim|x|→∞ f(x, t), uniformly in compact sets of R. We also assume that fi(∞, t) ∈ C1(R) holds.

We denote I and I∞ the energy functionals related to (S) and (S∞) respectively, with c(I) and c(I∞) being the

associated mountain pass level. We say that a weak solution (u, v) ∈ H \ (0, 0) is a ground state of System (S)

when I(u, v) ≤ I(u, v) for any other weak solution (u, v) ∈ H \ (0, 0), where H is a suitable Sobolev space.

2 Main Results

Theorem 2.1. Assume (A1)–(A3) and (H1)–(H6). If either c(I) < c(I∞) or I(u, v) ≤ I∞(u, v) hold, then System

(S) admits at least one ground state solution (u, v). Moreover, if I(u, v) ≤ I∞(u, v) then the ground state is at the

mountain pass level, that is, I(u, v) = c(I).

References

[1] A. Szulkin, T. Weth, ‘The method of Nehari manifold’, Handbook of Nonconvex Analysis and Applications,

Int. Press, Somerville, MA (2010), 597–632.

[2] J.C de Albuquerque, D. Ferraz, E. D. Silva, Ground states for fractional linear coupled systems via profile

decomposition, Nonlinearity, 34, (2021) 4787.

[3] F.O.V. Paiva, W. Kryszewski, A. Szulkin, Generalized Nehari manifold and semilinear Schrodinger equation

with weak monotonicity condition on the nonlinear term, Proc. Amer. Math. Soc. 145(11) (2017) 4783–4794.


REGULARIDADE DE INTERIOR PARA SOLUCOES DE EQUACOES FRACIONARIAS QUE

DEGENERAM COM O GRADIENTE

DISSON DOS PRAZERES1

1Universidade Federal de Sergipe, Brasil, [email protected]

Abstract

Nesta palestra iremos falar sobre regularidade interior para “viscosity solutions” de problemas de Dirichlet

nao-locais que degeneram quando o gradiente da soluc cao se anula. Apresentaremos estimativas Holder quando

a ordem da difusao e menor ou igual a 1 e estimativas Lipschitz quando a ordem da difusao e maior que 1. Alem

disso, no ultimo caso, discutiremos a possibilidade de obter estimativas Holder para o gradiente.

1 Introducao

Nesta apresentacao discutiremos sobre regularidade interior para “viscosity solutions” u de problemas elıpticos

nao-lineares da forma.

− |Du(x)|γI(u, x) = f(x) for x ∈ B1, (1)

onde γ > 0, f ∈ L∞(B1), Du(x) e o gradiente de u em x, e I(u, x) e um operador nao-local uniformemnte elıptico

da forma

I(u, x) = infi

supjIKij (u, x) (2)

onde

IKij(u, x) = P.V.

∫RN

[u(y) − u(x)]Kij(x− y)dy. (3)

Consideramos Kij : RN \ 0 → R uma familia de kernels simetricos tais que

λCσ,N

|x|N+σ≤ Kij ≤ Λ

Cσ,N

|x|N+σ, x = 0, (4)

onde σ ∈ (0, 2) and 0 < λ ≤ Λ <∞.

A principal dificuldade vem da presenca de |Du(x)|γ em (1), pois quando o gradiente de u vai para zero a

equacao degenera. Ou seja, a informacao que vem da equacao se perde quando o gradiente se anula.


Adaptamos para o nosso caso degenerado o metodo de Ishii-Lions nao-local a fim de obter o seguinte resultado,

Teorema 2.1. Sejam f ∈ L∞(B1) e I um operador da forma (2). Seja u ∈ L∞loc ∩ L1

σ uma “viscosity solution” do

problema (1). Entao,

Se 0 < σ < 1 entao u ∈ Cσ e

[u]Cσ(B1/2) ≤ C(∥u∥L1σ(RN ) + ∥u∥L∞(B1) + ∥f∥L∞(B1)).

Se σ = 1, entao u ∈ Cαpara todo α ∈ (0, 1), e

[u]Cα(B1/2) ≤ Cα(∥u∥L1σ(RN ) + ∥u∥L∞(B1) + ∥f∥L∞(B1)).

37

38

Se 1 < σ < 2 entaao u ∈ C0,1 e

[u]C0,1(B1/2) ≤ C0(∥u∥L1σ(RN ) + ∥u∥L∞(B1) + ∥f∥L∞(B1)),

e a constante C0 e uniformemente limitada quando σ → 2−.

Alem disto, quando σ esta proximo de 2 o problema (1) esta suficientemente proximo de um problema de

segunda ordem para o qual estimativas C1,α disponıvel e podemos obter o seguinte resultado,

Teorema 2.2. Sejam f ∈ L∞(B1) e I como em (2) definido a partir da familia de kernels Kijij, adicionalmente

satisfazendo a seguinte propriedade: existem um modulo de continuidade ω e um conjunto kiji,j ⊂ (λ,Λ) tal que

|Kij(x)|x|N+σ − kij | ≤ ω(|x|), |x| ≤ 1. (5)

Entao, existe um σ0 ∈ (1, 2) proximo de 2 tal que para σ0 < σ < 2 toda “viscosity solution” u para (1) e C1,α

para algum α ∈ (0, 1), e

[u]C1,α(B1/2) ≤ C0(∥u∥∞ + ∥f∥σ−11+γ∞ ).

References

[1] dos Prazeres, D and Topp E. - Interior regularity results for fractional elliptic equations that degenerate

with the gradient., J. Differential Equations 300 (2021), 814-829.


SUPERLINEAR FRACTIONAL ELLIPTIC PROBLEMS VIA THE NONLINEAR RAYLEIGH

QUOTIENT WITH TWO PARAMETERS

EDCARLOS D. SILVA1, M. L. M. CARVALHO2, M. L. SILVA3 & C. GOULART4

1 Universidade Federal de Goias, IME, Goiania-GO, Brazil, [email protected],2,

3 Universidade Federal de Goias, IME, Goiania-GO, Brazil, marcos leandro [email protected],4 Universidade Federal de Goias, IME, Goiania-GO, Brazil, [email protected]

Abstract

It is establish existence of weak solutions for nonlocal elliptic problems driven by the fractional Laplacian

where the nonlinearity is indefinite in sign. More specifically, we shall consider the following nonlocal elliptic

problem (−∆)su+ V (x)u = µa(x)|u|q−2u− λ|u|p−2u in RN ,

u ∈ Hs(RN ),

where s ∈ (0, 1), s < N/2, N ≥ 1 and µ, λ > 0. The potentials V, a : RN → R satisfy some extra assumptions.

The main feature is to find sharp parameters λ > 0 and µ > 0 where the Nehari method can be applied. In order

to do that we employ the nonlinear Rayleigh quotient together a fine analysis on the fibering maps associated

to the energy functional.

1 Introduction

In the present talk we shall consider nonlocal elliptic problems driven by the fractional Laplacian defined in the

whole space where the nonlinearity is superlinear at infinity and at the origin. Namely, we shall consider the

following nonlocal elliptic problem(−∆)su+ V (x)u = µa(x)|u|q−2u− λ|u|p−2u in RN ,

u ∈ Hs(RN ),(1)

where s ∈ (0, 1), s < N/2, N ≥ 1. Furthermore, we assume that 2 < q 0.

Assume also that V : RN → R is a continuous function and a : RN → R is nonnegative measurable function. It is

important to recall that the main difficult in order to consider weak solutions for Problem (1) comes from the fact

that the nonlinear term gλ,µ(x, t) = µa(x)|t|q−2t− λ|t|p−2t, x ∈ RN , t ∈ R is indefinite in sign. In fact, we observe

that

limt→0

gλ,µ(x, t)

t= 0, lim

t→∞

gλ,µ(x, t)

t= −∞ (2)

and gλ,µ(x, t) > 0 for each t ∈ (0, δ), x ∈ RN for some δ > 0. Hence, we obtain that gλ,µ(x, t) is a sign changing

nonlinearity. Semilinear elliptic problems have widely considered in the last years since the seminal work [1].

2 Main Results

As was told in the introduction we shall consider existence and nonexistence of nontrivial weak solutions for the

Problem (1) looking for the parameters λ > 0 and µ > 0. The main idea here is to ensure sharp conditions on the

parameters λ and µ such that the Nehari method together with the nonlinear Rayleigh quotient can be applied, see

[2, 3]. Throughout this work we assume the following assumptions:

39

40

(Q) It holds µ, λ > 0 and 2 < q 0 for all x ∈ RN ;

(V1) For each M > 0 it holds that |x ∈ RNn : V (x) ≤M| < +∞.

(a0) It holds that a ∈ L∞(RN ) where a(x) > 0 a. e. in x ∈ RN .

It is important to mention that the working space for our work is defined by

X =

v ∈ Hs(RN ) :

∫V (x)v2dx < +∞

.

Notice that X is a Hilbert space. It is worthwhile to emphasize that that the energy functional Eλ,µ : X → Rassociated to Problem (1) is given by

Eλ,µ(u) =1

2||u||2 − µ

q∥u∥qq,a +

λ

p∥u∥pp, u ∈ X, (1)

where

∥u∥qq,a =

∫a(x)|u|qdx and ∥u∥pp =

∫|u|pdx, u ∈ X.

Under our hypotheses we observe that Eλ,µ belongs to C2(X,R) for each λ > 0 and µ > 0. Moreover, a function

u ∈ X is a critical point for the functional Eλ,µ if and only if u is a weak solution to the elliptic Problem (1). Now,

by using the same ideas introduced we shall consider the Nehari method for our main Problem (1). As a product,

we shall state our first main result as follows:

Theorem 2.1. Suppose (Q), (V0) − (V1) and (a0). Then for each λ > 0 we obtain that 0 < µn < µe < ∞.

Furthermore, there exists λ∗ > 0 such that for each µ > µn Problem (1) admits at least a weak solution uλ,µ ∈ X

whenever λ ∈ (0, λ∗) which it satisfies the following assertions: E′′λ(uλ,µ)(uλ,µ, uλ,µ) < 0 and there exists Dµ > 0

such that Eλ,µ(uλ,µ) ≥ Dµ and uλ,µ → 0 in X as µ→ ∞.

Now we assume the following hypothesis:

(a1) It holds that a ∈ L∞(RN ) ∩ Lr(RN ) with r = (p/q)′ = p/(p− q) and a(x) > 0 a.e. in x ∈ RN .

Hence, we can written our next main result in the following form:

Theorem 2.2. Suppose (Q), (V0) − (V1) and (a1). Then for each λ > 0 we obtain that 0 < µn < µe < ∞.

Furthermore, there exits λ∗ > 0 such that for each µ > µn Problem (1) admits at least a ground state solution

vλ,µ ∈ X taking into account one of the following conditions: µ ∈ [µe,∞), λ > 0 and µ ∈ (µn, µe), λ ∈ (0, λ∗).

Moreover, the weak solution vλ,µ satisfies the following assertions: It holds that E′′λ(vλ,µ)(vλ,µ, vλ,µ) > 0. Moreover,

∥vλ,µ∥ → ∞ in X as µ → ∞. For each µ ∈ (µn, µe) we obtain that Eλ,µ(vλ,µ) > 0. For µ = µe it follows that

Eλ(vλ,µ) = 0. For each µ > µe we obtain also that Eλ(vλ,µ) < 0.

References

[1] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in: CBMS

Regional Conference Series in Mathematics, vol.65, Published for the Conference Board of the Mathematical

Sciences, Washington, DC, 1986, by the American Mathematical Society, Providence, RI.

[2] Y. Il’yasov, On extreme values of Nahari manifold method via nonlinear Rayleigh’s quotient, Topol. Methods

Nonlinear Anal. 49 (2017), no. 2, 683-714.

[3] Y. Il’yasov, On nonlocal existence results for elliptic equations with convex-concave nonlinearities, Nonl. Anal.:

Th., Meth. Appl., 61(1-2),(2005) 211-236.


EXISTENCE OF SOLUTIONS FOR A FRACTIONAL CHOQUARD–TYPE EQUATION IN R WITH

CRITICAL EXPONENTIAL GROWTH

EUDES MENDES BARBOZA1, RODRIGO CLEMENTE2 & JOSE CARLOS DE ALBUQUERQUE3

1Departamento de de Matematica, UFRPE, PE, Brasil, [email protected],2Departamento de de Matematica, UFRPE, PE, Brasil, [email protected],3Departamento de de Matematica, UFPE, PE, Brasil, [email protected]

Abstract

In this work we study the following class of fractional Choquard–type equations

(−∆)1/2u+ u =(Iµ ∗ F (u)

)f(u), x ∈ R,

where (−∆)1/2 denotes the 1/2–Laplacian operator, Iµ is the Riesz potential with 0 < µ < 1 and F is the

primitive function of f . We use Variational Methods and minimax estimates to study the existence of solutions

when f has critical exponential growth in the sense of Trudinger–Moser inequality.

1 Introduction

This talk is based on [4], here we are concerned with existence of solutions for a class of fractional Choquard–type

equations

(−∆)su+ u =(Iµ ∗ F (u)

)f(u), x ∈ RN , (1)

where (−∆)s denotes the fractional Laplacian, 0 < s < 1, 0 < µ < N , F is the primitive function of f ,

Iµ : RN\0 → R is the Riesz potential defined by

Iµ(x) := Aµ1

|x|N−µ, where Aµ :=

Γ

(N − µ

2

)Γ(µ

2

)π

N2 2µ

,

and Γ denotes the Gamma function. We consider the “limit case” when N = 1, s = 1/2 and a Choquard–

type nonlinearity with critical exponential growth motivated by a class of Trudinger–Moser inequality. The main

difficulty is to overcome the “lack of compactness” inherent to problems defined on unbounded domains or involving

nonlinearities with critical growth. In order to apply properly the Variational Methods, we control the minimax

level with fine estimates involving Moser functions (see [7]), but here in the context of fractional Choquard–type

equation.

Nonlinear elliptic equations involving nonlocal operators have been widely studied both from a pure

mathematical point of view and their concrete applications, since they naturally arise in many different contexts,

such as, among the others, obstacle problems, flame propagation, minimal surfaces, conservation laws, financial

market, optimization, crystal dislocation, phase transition and water waves, see for instance [2, 3] and references

therein.

Inspired by [1], our goal is to establish a link between Choquard–type equations, 1/2– fractional Laplacian and

nonlinearity with critical exponential growth. We are interested in the following class of problems

(−∆)1/2u+ u =(Iµ ∗ F (u)

)f(u), x ∈ R, (P)

41

42

where F is the primitive of f . In order to use a variational approach, the maximal growth is motivated by the

Trudinger–Moser inequality first given by T. Ozawa [6] and later extended by S. Iula, A. Maalaoui, L. Martinazzi

[5]. Precisely, it holds

supu∈H1/2(R)

∥(−∆)1/4u∥2≤1

∫R

(eαu2

− 1) dx

<∞, α ≤ π,

= ∞, α > π.

In this work we suppose that f : R → R is a continuous function satisfying the following hypotheses:

(f1) f(t) = 0, for all t ≤ 0 and 0 ≤ f(t) ≤ Ceπt2

, for all t ≥ 0;

(f2) There exist t0, C0 > 0 and a ∈ (0, 1] such that 0 < taF (t) ≤ C0f(t), for all t ≥ t0;

(f3) There exist p > 1 − µ and Cp = C(p) > 0 such that f(t) ∼ Cptp, as t→ 0;

(f4) There exists K > 1 such that KF (t) < f(t)t for all t > 0, where F (t) =∫ t

0f(τ) dτ ;

(f5) lim inft→+∞

F (t)

eπt2=√β0 with β0 > 0.

2 Main Results

We are in condition to state our main result:

Theorem 2.1. Suppose that 0 < µ < 1 and assumptions (f1) − (f5) hold. Then, Problem (P) has a nontrivial

weak solution.

Remark 2.1. Though there have been many works on the existence of solutions for problem (1), as far as we

know, this is the first work considering a fractional Choquard–type equation involving 1/2–Laplacian operator and

nonlinearity with critical exponential growth. Particularly, our Theorem 2.1 is a version of Theorem 1.3 of [1] for

1/2-Laplacian operator.

References

[1] alves, c. o., cassani, d., tarsi, c., yang, m.- Existence and concentration of ground state solutions for a

critical nonlocal Schrodinger equation in R2, J. Differential Equations 261, no. 3, 1933–1972, 2016.

[2] caffarelli, l.- Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symp.

7, Springer, Heidelberg, 37–52, 2012.

[3] di nezza, e., palatucci, g., valdinoci, e.- Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci.

Math. 136, 521-573, 2012.

[4] genuino, r. c., albuqquerque,j.c., barboza, e.-Existence of solutions for a fractional Choquard-type

equation in R with critical exponential growth. Zeitschrift Fur Angewandte Mathematik Und Physik, 72, p. 16,

2021.

[5] iula, S., maalaoui, a., martinazzi, l.-A fractional Moser-Trudinger type inequality in one dimension and

its critical points, Differential Integral Equations 29, no. 5/6, 455–492, 2016.

[6] Ozawa, T.- On critical cases of Sobolev’s inequalities, J. Funct. Anal. 127, 259–269, 1995.

[7] takahashi, f.- Critical and subcritical fractional Trudinger-Moser-type inequalities on R. Advances in

Nonlinear Analysis 1, 868–884, 2019.


CRITICAL METRICS OF THE SK OPERATOR

FLAVIO A. LEMOS1 & EZEQUIEL R. BARBOSA2

1Departamento de Matematica, UFOP, MG, Brasil, [email protected],2Departamento de Matematica, Universidade Federal de Minas Gerais, UFMG, MG, Brasil, [email protected]

Abstract

Given a smooth compact Riemannian n-manifold (M, g) with positive scalar curvature, we prove that any

complete critical metric of the Lk-norm of the scalar curvature, has constant scalar curvature.

1 Introduction

Let (Mn, g), n ≥ 3, be a n-dimensional smooth Riemannian manifold and consider the functional

Sk(g) =

∫M

RkdVg (1)

on the space of Riemannian metrics on Mn, where k ∈ N, Rg and dVg denote the scalar curvature and the volume

for of g respectively. In the case k = 2, Giovanni Catino [4] proved the following theorem

Theorem 1.1. Let (Mn, g), n ≥ 3, be a complete critical metric of S2 with positive scalar curvature. Then (Mn, g)

has constant scalar curvature.

Urging for a more general result, we calculated the first variation of Sk(g), using derivatives formulas (see [3])

in the direction of h (g(t) = g + th)

δSk(g)[h] =

∫M

(kRk−1δR+1

2Rktr(h))dVg

=

∫M

(−kRk−1∆gtr(h) + kRk−1div2(h) − kRk−1 < Ric, h >g +1

2Rktr(h))dVg

=

∫M

(−k∆gRk−1g + k∇2

gRk−1 − kRk−1Ric+

1

2Rkg)hdVg.

Remark 1.1. We take h with compact support, such that we can apply the Divergence Theorem for (2, 0)-tensor.

Hence, the Euler Lagrange equation for a critical metric of Sk in the direction of h is given by

Rk−1Ric−∇2g(Rk−1) + ∆g(Rk−1)g =

1

2

Rk

kg. (2)

By induction, we can proof that

∇2g(Rk−1)(X,Y ) = (k − 1)Rk−2∇2

gR(X,Y ) + (k − 1)(k − 2)Rk−3X(R)Y (R). (3)

By (2) and (3)

∆gR =

((n− 2k)

2k(n− 1)(k − 1)

)R2 − (k − 2)

|∇gR|2

R(4)

By above equalities; any critical metric of Sk is scalar flat if n is odd, whereas it is either scalar flat or Einstein if

n = 2k.

In this paper we will focus on complete critical metrics of Sk. As for as we know, complete critical metrics of

Sk were not studied yet. Our main result characterizes critical metrics with positive scalar curvature.

43

44

Theorem 1.2. Let (Mn, g), n ≥ 3, be a complete critical metric of Sk with positive scalar curvature and k ≥ 2 ∈ N.Then (Mn, g) has constant scalar curvature.

Theorem 1.3. Let (Mn, g), n ≥ 3, be a complete critical metric of Sk with positive scalar curvature and k ≥ 2 ∈ N.If n < 2k, then (Mn, g) is scalar flat.

In particular, from equations (2) and (3), if n = 2k , there are no complete critical metrics of Sk with positive

scalar curvature, whereas, every complete 2k-dimensional critical metric Sk with positive scalar curvature is either

flat or Einstein with positive scalar curvature.

References

[1] M. T. Anderson - Extrema of curvature functionals on the space of metrics on 3-manifolds,, Cal. Var. PDEs 5

(1997), 199-269.

[2] M. T. Anderson - Extrema of curvature functionals on the space of metrics on 3-manifolds, II, Cal. Var. PDEs

12 (2001), 1-58.

[3] Bennett Chow, Peng Lu and Lei Ni - Hamilton’s Ricci Flow, Lectures in Contemporary Mathematics, Science

Press, Beijing (2006).

[4] Giovanni Catino - Critical Metrics of the L2-Norm of the Scalar Curvature, Proc. Amer. Math. Soc. 142 (2014),

3981-3986.

[5] G. Wei and W. Wylie - Comparison geometry for the Bakry-Emery Ricc tensor, J. Differential Geom. 83 (2009),

no. 2, 377-405.


EXISTENCE OF SOLUTION FOR IMPLICIT ELLIPTIC EQUATIONS INVOLVING THE

P-LAPLACE OPERATOR

GABRIEL RODRIGUEZ V.1, EUGENIO CABANILLAS L.2, WILLY BARAHONA M.3, LUIS MACHA C.4 & VICTOR

CARRERA B.5

1Instituto de Investigacion, FCM,UNMSM, Peru, [email protected],2Instituto de Investigacion, FCM,UNMSM, Peru, [email protected],3Instituto de Investigacion, FCM,UNMSM, Peru, [email protected],

4Instituto de Investigacion, FCM,UNMSM, Peru, [email protected],5Instituto de Investigacion, FCM,UNMSM, Peru, [email protected]

Abstract

In this research we will study the existence of weak solutions for a class of implicit elliptic equations involving

the p-Laplace Operator. Using a Krasnoselskii-Schaefer type theorem we establish our result.

1 Introduction

In this article we focus on the following boundary problem

−∆pu = f(x, u,∇u,∆pu) + g(x, u,∇u)|u|ts in Ω

u = 0 on Γ(1)

where Ω is a bounded domain with smooth boundary Γ in Rn(n ≥ 3) , f : Ω×R×Rn×R → R, g : Ω×R×Rn → R,

2 ≤ p < +∞ and s > 1, t ≥ 0.

Implicit elliptic equations have been intensively studied in the literature, see for example [1, 3]. More recently

Precup [4] studied the case p = 2 , t = 0. Motivated by the above works, we are devoted to study problem (1.1).

2 Assumptions and Main Results

We give the following hypotheses.

(A1) There exist a, b, c ≥ 0 such that

|f(x, y, z, w) − f(x, y, z, w)| ≤ a|y − y| + b|z − z|p−1 + c|w − w| , f(· , 0, 0, 0) ∈ Lp(Ω)

(A2) There exist constants a0, b0 ≥ 0, α ∈ [1, p∗/(p∗)′], β ∈ [1, p/(p∗)′]; and h ∈ Lp(Ω) such that

|g(x, y, z)| ≤ a0|y|α + b0|z|β + h(x) , ∀y ∈ R, z ∈ Rn and a.e x ∈ Ω

(A3) yg(x, y, z) ≤ σ|y|p , ∀y ∈ R, z ∈ Rn a.e x ∈ Ω, for some σ < σ0λ1 , 0 < σ0 < 1 , λ1 is the first

eingenvalue of (−∆p,W1,p0 (Ω))

(A4) ℓ :=a

λ1+

b

λ1,p1

+ c < 1 , G0 = 1 − ℓ

Our main result is the following theorem

Theorem 2.1. Let (p∗)′ ≤ τ ≤ p. Suppose (A1) - (A4) hold. Then (1.1) has at least one weak solution u ∈W 1,p0 (Ω)

with ∆pu ∈ Lτ (Ω).

45

46

Proof We transform (1.1) into an equivalent problem of fixed point, where the associated operator is a sum of a

contraction with a completely continuous mapping. Then, we apply a result in [2].

References

[1] Bonanno G., Marano S., - Elliptic problems in Rn with discontinuous nonlinearities, Proc. Edinbungh

Math. Soc. 43(2000) 545-558.

[2] Burton T.A., Kirk C., - A fixed point theorem of Krasnoselskii-Schaefer type, Math Nachr., 189(1998) 23-31.

[3] Carl S. , Hoikkila S., - Discontinuous implicit elliptic boundary problems , Diffrential Integral Eq. 11(1999)

823-834.

[4] Precup R., - Implicit elliptic equations via Krasnoselskii - Schaefer type theorems , EJQDE, 87(2020) 1-9.


UM SISTEMA DE TIPO SCHRODINGER-BORN-INFELD

GAETANO SICILIANO1,†

1Universidade de Sao Paulo, Instituto de Matematica e Estatıstica, Sao Paulo, [email protected]

Abstract

Apresentamos um sistema envolvendo a equacao de Schrodinger nao linear e a equacao da electrostatica

de Born-Infeld e procuramos solucoes no caso radial em R3. Dependendo do parametro p da nao linearidade

tecnicas diferentes sao usadas para mostrar a existencia de solucoes.

1 Introducao e resultado principal

Consideramos o seguinte sistema nao linear de tipo Schrodinger-Born-Infeld−∆u+ u+ ϕu = |u|p−1u in R3,

−div

(∇ϕ√

1 − |∇ϕ|2

)= u2 in R3,

(1)

com p dado e nas incognitas u, ϕ : R3 → R.

Esse sistema aparece na busca de solucoes estacionarias da equacao de Schrodinger acoplada com a teoria

eletromagnetica de Born-Infeld, em lugar da classica teoria de Maxwell. A vantagem dessa nova teoria e que

elimina o problema da energia infinita que a Teoria de Maxwell associa a uma carga puntual. De fato, na Teoria

de Maxwell a busca de ondas estacionarias leva ao sistema−∆u+ u+ ϕu = |u|p−1u in R3,

−∆ϕ = u2 in R3,

(2)

e e facil de ver que a solucao fundamental Φ do Laplaciano satisfaz∫R3 |∇Φ|2 = +∞, ou seja a energia asociada a

uma carica puntual e infinita.

Contudo a desvantagem da electrodinamica de Born-Infeld e que a equacao do campo electrico, ou seja a segunda

em (1), e nao linear, quando na teoria de Maxwell por ser a equacao de Poisson e muito mais simples.

Em dois trabalhos distintos nos provamos existencia de solucoes para o sistema (1) que e muito menos estudado

do sistema (2). Nos usamos Metodos Variaciones, Teoria do Ponto Crıtico e oportunas perturbacoes no sistema.

Em particular, em [1] com A. Azzollini (Universita della Basilicata, IT) and A. Pomponio (Politecnico di Bari,

IT) mostramos o seguinte resultado.

Teorema 1.1. Pora cada p ∈ (5/2, 5), o problem (1) possui uma solucao radial de ground state, ou seja uma

solucao (u, ϕ) que minimiza o funcional da acao entre todas as outras solucoes.

No trabalho [3] com Z. Liu (China University of Geosciences) mostramos a existencia do ground state tambem

por valores menores de p cobrindo o caso p ∈ (2, 5/2]. Alem disso, provamos o resultado em presencia de uma nao

linearidade com crescimento crıtico e abordamos o problema da multiplicidade de solucoes encontrando infinitas

solucoes com nıveis de energia que tende para +∞.

47

48

Destacamos que no trabalho [1] para garantir a geometria do Paso da Montanha foi usado o “monotonicity

trick” de Jeanjean, que consiste em introduzir um parametro de controle multiplicativo λ em um termo ja presente

na equacao e mostrar que quando λ tende para 1 se obtem uma solucao do sistema inicial. Por outro lado essa

tecnica nao funciona por valores menores de p. Para contornar essa dificuldade, em [3] usamos um diferente metodo

de perturbacao que consiste em adicionar dessa vez na equacao um termo contendo o parametro de controle λ

e mandar o λ para 0. Nesse caso as contas sao bem mais envolvidas mas mesmo assim conseguimos mostrar a

geometria do Passo da Montanha e a condicao de compacidade necessaria para obter existencia de solucoes.

References

[1] azzollini, a., pomponio a. and siciliano, g. - On the Schrodinger-Born-Infeld System. Bull Braz Math

Soc, New Series, DOI 10.1007/s00574-018-0111-y.

[2] siciliano, g. and liu, z. - A perturbation approach for the Schrodinger-Born-Infeld system: Solutions in the

subcritical and critical case. J. Math. Anal. Appl., 503, 125326, 2021.


EQUACOES DE SCHRODINGER QUASELINEARES COM POTENCIAIS SINGULARES E SE

ANULANDO ENVOLVENDO NAO LINEARIDADES COM CRESCIMENTO CRITICO

EXPONENCIAL

GILSON M. DE CARVALHO1, YANE L. R. ARAUJO2 & RODRIGO G. CLEMENTE3

1Departamento de Matematica, UFRPE, PE, Brasil, [email protected],2Departamento de Matematica, UFRPE, PE, Brasil, [email protected],

3Departamento de Matematica, UFRPE, PE, Brasil, [email protected]

Abstract

Neste trabalho nos estudamos e estabelecemos resultados de existencia de solucao fraca e de nao existencia

de solucao classica para a seguinte classe de equacoes de Schrodinger

−∆Nu+ V (|x|)|u|N−2u = Q(|x|)h(u) em RN ,

em que N ≥ 2, V e Q sao potenciais contınuos que podem ser ilimitados na origem ou se anularem no infinito e h

e uma nao linearidade que possui um crescimento crıtico exponencial com respeito a desigualdade de Trudinger-

Moser. Para atingirmos os nossos objetivos atacamos o problema usando uma abordagem variacional, bem como

fizemos uso de uma desigualdade do tipo Trudinger-Moser e de resultados do tipo princıpio da criticalidade

simetrica.

1 Introducao

Aqui estamos interessados em estabelecer resultados de existencia e de nao existencia de solucao para a seguinte

classe de problemas −∆Nu+ V (|x|)|u|N−2u = Q(|x|)h(u), se x ∈ RN

u(x) → 0, quando |x| → +∞,(P)

em que N ≥ 2 e ∆Nu = div(|∇u|N−2∇u) denota o operador N -Laplaciano da funcao u. Primeiramente, para o

estudo de existencia de solucao fraca, vamos considerar V e Q potenciais contınuos satisfazendo:

(V1) V : (0,+∞) → R, V (r) > 0 para todo r > 0 e existem constantes a > −N e a0 > −N tais que

0 < lim infr→0+

V (r)

rb0e 0 < lim inf

r→+∞

V (r)

ra.

(Q1) Q : (0,+∞) → R, Q(r) > 0 para todo r > 0 e existem constantes b0 > −N e b < a tais que

lim supr→0+

Q(r)

ra0< +∞ e lim sup

r→+∞

Q(r)

rb< +∞.

Tambem pedimos que a nao linearidade h : R → R seja contınua e satisfaz:

(H1) Existe α0 > 0 tal que

lims→+∞

h(s)

eαsN/(N−1)=

0, ∀ α > α0

+∞, ∀ α < α0.

(H2) lims→0

h(s)

sN−1= 0;

49

50

(H3) Existe µ > N tal que

0 ≤ µH(s) := µ

∫ s

0

h(t) dt ≤ sh(s) para todo s ∈ R \ 0 ;

(H4) Existem ξ > 0 e κ > N tais que

H(s) ≥ ξsκ, ∀ s ≥ 0;

(H5) h(s)/sN e nao decrescente para s > 0.

Assumindo tais hipoteses, definindo um espaco adequado, usando um resultado de imersao, uma desigualdade do

tipo Trudinger-Moser, metodos variacionais e um resultado do tipo princıpio da criticalidade simetrica temos o

seguinte resultado.

Teorema 1.1. Suponha que V e Q sao potenciais satisfazendo (V1) e (Q1), respectivamente, e que h e uma nao

linearidade obedecendo as condicoes (H1) − (H4), entao (P) possui uma solucao fraca nao nula e nao negativa.

Alem disso, se h tambem satisfaz (H5), temos que (P) admite uma solucao ground state.

Por outro lado, se assumirmos V , Q e h satisfazendo

(V ) V : (0,+∞) → R e contınuo, V (r) ≥ 0 e existe a ∈ R tal que

lim supr→+∞

V (r)

ra< +∞;

(Q) Q : (0,+∞) → R e contınuo, Q(r) ≥ 0 e existe b ≥ a tal que

lim infr→+∞

Q(r)

rb> 0;

(h) h : R → R e contınuo e existem ξ > 0 e p ≤ N − 1 tais que

h(s) ≥ ξsp, para todo s > 0;

respectivamente, fazendo uso da formulacao em coordenadas radiais do N -Laplaciano e usando argumentos de

contradicao obtemos nosso resultado de nao existencia de solucao.

Teorema 1.2. Assuma que as condicoes (V ), (Q) e (h) sao satisfeitas. Entao, o problema (P) nao possui uma

solucao classica radial e positiva.

References

[1] Araujo, Y.L.; de Carvalho, G.M. and Clemente, R.G. - Quasilinear Schrodinger equations with singular

and vanishing potentials involving nonlinearities with critical exponential growth. - Topol. Methods Nonlinear

Anal., 57 NA° 1, (2021) 317-342.

[2] F. Albuquerque, C. Alves, E. Medeiros - Nonlinear Schrodinger equation with unbounded or decaying

radial potentials involving exponential critical growth in R2. - J. Math. Anal. Appl., 409 (2014) 1021-1031.

[3] J. Su, Z-Q. Wang, M. Willem - Weighted Sobolev embedding with unbounded and decaying radial

potentials. J. Differential Equations, 238 (2007), no. 1, 201-219.


THE LIMITING BEHAVIOR OF GLOBAL MINIMIZERS IN NON-REFLEXIVE ORLICZ-SOBOLEV

SPACES

GREY ERCOLE1, GIOVANY M. FIGUEIREDO2, VIVIANE M. MAGALHAES3 & GILBERTO A. PEREIRA4

1Departamento de Matematica, UFMG, MG, Brasil, [email protected],2Departamento de Matematica, UnB, DF, Brasil, [email protected],

3Departamento de Matematica, UFMG, MG, Brasil, [email protected],4Departamento de Matematica, UFOP, MG, Brasil, [email protected]

Abstract

Let Ω be a smooth, bounded N -dimensional domain. For each p > N, let Φp be an N-function satisfying

pΦp(t) ≤ tΦ′p(t) for all t > 0, and let Ip be the energy functional associated with the equation −∆Φpu = f(u)

in the Orlicz-Sobolev space W1,Φp

0 (Ω). We prove that Ip admits at least one global, nonnegative minimizer up

which, as p→ ∞, converges uniformly on Ω to the distance function to the boundary ∂Ω.

1 Introduction

Let Ω be a smooth, bounded N -dimensional domain and denote by dΩ the distance function to the boundary ∂Ω,

defined by

dΩ(x) := infy∈∂Ω

|x− y| , x ∈ Ω.

For each p > N, let ϕp : [0,∞) → [0,∞) be an increasing function of class C1 such that

pΦp(t) ≤ tΦ′p(t) for all t > 0,

where Φp : R → [0,∞) is the N-function defined by

Φp(t) :=

∫ t

0

sϕp(|s|)ds, t ∈ R.

Let f : R → R be a continuous function enjoying the following properties:

(f1) f(−t) + f(t) ≥ 0 for all t ≥ 0,

(f2) F, the primitive of f given by F (t) =∫ t

0f(s)ds, is strictly increasing on [0, ∥dΩ∥∞], and

(f3) there exist constants a, b, r and t0, with a ≥ 0, b > 0 and r, t0 ≥ 1, such that

0 ≤ f(t) ≤ a+ btr−1 for all t ≥ t0.

Let W1,Φp

0 (Ω) be the Orlicz-Sobolev space generated by Φp and consider the energy functional

Ip(u) :=

∫Ω

Φp(|∇u|)dx−∫Ω

F (u)dx, u ∈W1,Φp

0 (Ω),

associated with the Dirichlet problem−div(ϕp(|∇u|)∇u) = f(u) in Ω

u = 0 on ∂Ω.

51

52

Under the above hypotheses on ϕp, the N-function Φp may grow at infinity faster than any polynomial (see [2]).

If this is the case, Φp does not satisfy the ∆2-condition and, consequently, neither W1,Φp

0 (Ω) is reflexive nor its

modular functional u 7→∫Ω

Φp(|∇u|)dx is of class C1.

Considering these facts and taking into account that the modular functional is always convex and sequentially

lower semicontinuous with respect to the weak-star topology (see [6]) we adopt in this paper the following definition,

according to [5]: a function u ∈W1,Φp

0 (Ω) is a critical point of Ip if∫Ω

Φp(|∇u|)dx <∞ and the variational inequality∫Ω

Φp(|∇v|)dx−∫Ω

Φp(|∇u|)dx ≥∫Ω

f(u)(v − u)dx (1)

holds for all v ∈W1,Φp

0 (Ω).

We show that Ip admits at least one global, nonnegative minimizer up and, under the additional assumptions

limp→∞

Φp(1) = 0 and limp→∞

(Φp(1))1p = 1,

we prove that up converges uniformly on Ω to dΩ, as p→ ∞. This convergence result generalizes the corresponding

ones of [1, 2, 5].

References

[1] bocea, m. and mihailescu, m. - On a family of inhomogeneous torsional creep problems. Proc. Amer. Math.

Soc., 145, 4397-4409, 2017.

[2] farcaseanu, m. and mihailescu, m. - On a family of torsional creep problems involving rapidly growing

operators in divergence form. Proc. Roy. Soc. Edinburgh Sect. A, 149, 495-510, 2019.

[3] kawohl, b. - On a family of torsional creep problems. J. Reine Angew. Math., 410, 1-22, 1990.

[4] Le, v.k. and Schmitt, k. - Quasilinear elliptic equations and inequalities with rapidly growing coefficients.

J. London Math. Soc., 62, 852-872, 2000.

[5] szulkin, a. - Minimax principles for lower semicontinuous functions and applications to nonlinear boundary

value problems. Ann. Inst. H. Poincare Anal. Non Lineaire, 3, 77-109, 1986.


VARIATIONAL FREE TRANSMISSION PROBLEMS OF BERNOULLI TYPE

HARISH SHRIVASTAVA1 & DIEGO MOREIRA2

1Universidade Federal do Ceara,2Universidade Federal do Ceara

Abstract

We study functionals of the following type

JA,f,Q(v) :=

∫Ω

A(x, u)|∇u|2 − f(x, u)u+Q(x)λ(u) dx

here A(x, u) = A+(x)χu>0+A−(x)χu<0, f(x, u) = f+(x)χu>0+f−(x)χu<0 and λ(x, u) = λ+(x)χu>0+

λ−(x)χu≤0. We assume 0 < λ− < λ+ < ∞ and 0 ≤ Q ≤ q2. We prove the optimal regularity (C0,1−) of

minimizers of the functional indicated above when coefficients A± are continuous functions with µ ≤ A± ≤ 1µ,

f ∈ LN (Ω) and Q is bounded and continuous.

1 Introduction

In various applied sciences, many phenomenas are modelled by transmission problem also known as phase

transmission problems. These kind of models naturally appear when we study the diffusion of a quantity through

different media.

Let us look at example of the stationary state of the ice-water combination and studying the diffuson of heat

(related to the temperature T ) T : Ω → RN , Ω being the domain under study. We can say that in ice the

diffusion is determined by an operator corresponding to solid state and in water, the diffusion is determined by

an operator corresponging to liquid state. As a combination, above mentioned phenomena can be posed in the

following variational setup, ∫Ω

⟨A(x)∇v,∇v⟩ − f(x, v)v + γ(x, v) dx (1)

with

A(x, v) := A+(x)χv>0 +A−(x)χv≤0, f(x, v) := f+(x)χv>0 + f−(x)χv≤0, γ(x, v) := γ+(x)χv>0 + γ−(x)χv≤0

The matrices A± satisfy the ellipticity condition for any ξ ∈ RN λ|ξ|2 ≤ ⟨A±ξ, ξ⟩ ≤ Λ|ξ|2, f± ∈ LN (Ω), γ± ∈ C(Ω).

This class of problems has attracted a lot of attention in recent years. For example, the [2] consider the PDE

with jump in leading coefficients and show Lipschitz regularity of solution when A± ∈ Cα(Ω). Furthermore, [1] and

[3] consider the variational setup as in (1). These works prove that the regularity of minimizers tend to C0,1− as

the jump of the coefficients A+ and A− tend to zero. In our work, we are successful in removing the small jump

condition and prove that minimizers are C0,1− regular independent of quantity of jump |A+ −A−| in any norm.

2 Main Results

2.1 Regularity of Elliptic PDE with continuous coefficients

We are concerned about the regularity of (weak) solutions to the following PDE in B1

div(A(x)∇u) = f. (2)

where A ∈ C(Ω)N×N (B1) and f ∈ LN (B1).

53

54

Proposition 2.1. Suppose u ∈ H1(B1) ∩ L∞(B1) be a weak solution to (2) in B1. Then for any 0 < α < 1, we

have u ∈ Cα(B1/2) with the following estimates

[u]C0,α(B1/2) ≤ C(N,α, µ, ωa,B3/4)(∥u∥L∞(B1) + ∥f∥LN (B1)

). (3)

Here ωA,B3/4is uniform the modulus of continuity of A in B3/4. In particular, we have

∥u∥Cα(B1/2) ≤ C(N,α, µ, ωa,B3/4)(∥u∥L∞(B1) + ∥f∥LN (B1)

). (4)

2.2 Optimal regularity of minimizers

Theorem 2.1. Suppose u ∈ H1(Ω) be a minimizer of JA,f,Q(·,Ω). Then u is locally bounded in Ω and for every

α ∈ (0, 1) and x0 ∈ Ω we have

∥u∥∗Cα(Br(x0))≤ C(N, p, α, µ, q2, λ+, ωA±,B2r(x0))

(r + ∥u∥L∞(B2r(x0)) + r∥f∥LN (B2r(x0))

).

Here r < d4 (d := dist(x0, ∂Ω)), ωA±,B2r(x0) is the modulus of continuity of A± in the ball B2r(x0). In particular,

[u]Cα(Br(x0)) ≤C(N, p, α, µ, q2, λ+, ωA±,B2r(x0))

rα(r + ∥u∥L∞(B2r(x0)) + r∥f∥LN (B2r(x0))

).

References

[1] M. D. Amaral and E. Teixeira, Free transmission problems, Communications in Mathematical Physics, 337

(2015), pp. 1465–1489.

[2] S. Kin, K.-A. Lee, and H. Shahgholian, An elliptic free boundary arising from the jump of conductivity,

Nonlinear Analysis, 161 (2017), pp. 1–29.

[3] H. Shrivastava, A non-isotropic free transmission problem governed by quasi-linear operators, Ann. Mat. Pur.

Appl., (2021).


COMPACTNESS WITHIN THE SPACE OF COMPLETE, CONSTANT Q-CURVATURE METRICS

ON THE SPHERE WITH ISOLATED SINGULARITIES

JOAO HENRIQUE ANDRADE1, JOAO MARCOS DO O2 & JESSE RATZKIN3

1Instituto de Matematica e Estatıstica, USP, SP, Brasil, [email protected],2Departamento de Matematica, UFPB, PB, Brasil, [email protected],

3Departamento de Matematica, JMU, WU, Alemanha, [email protected]

Abstract

In this paper we consider the moduli space of complete, conformally flat metrics metrics on a sphere with

k punctures having constant positive Q-curvature and positive scalar curvature. Previous work has shown that

such metrics admit an asymptotic expansion near each puncture, allowing one to define an asymptotic necksize

of each singular point. We prove that any set in the moduli space such that the distances between distinct

punctures and the asymptotic necksizes all remain bounded away from zero is sequentially compact, mirroring a

theorem of D. Pollack about singular Yamabe metrics. Along the way we define a radial Pohozaev invariant at

each puncture and refine some a priori bounds of the conformal factor, which may be of independent interest.

1 Introduction

In recent years many people have pursued parts of Yamabe’s program for other notions of curvature. In the present

note, we explore a part of the singular Yamabe program as applied to the fourth order Q-curvature, which is a

higher order analog of scalar curvature. On a Riemannian manifold (M, g) of dimension n ≥ 5, the Q-curvature is

Qg = − 1

2(n− 1)∆gRg −

2

(n− 2)2|Ricg|2 +

n3 − 4n2 + 16n− 16

8(n− 1)2(n− 2)2R2

g, (1)

where Rg is the scalar curvature of g, Ricg is the Ricci curvature of g, and ∆g is the Laplace–Beltrami operator of

g. After a conformal change, the Q-curvature transforms as

g = u4

n−4 g → Qg =2

n− 4u−

n+4n−4Pgu, (2)

where Pg is the Paneitz operator

Pgu = ∆2gu+ div

(4

n− 2Ricg(∇u, ·) − (n− 2)2 + 4

2(n− 1)(n− 2)Rg⟨∇u, ·⟩

)+n− 4

2Qgu. (3)

The Q-curvature of the round metricg is n(n2−4)

8 , and setting Qg to be this value gives the equation

Pgu =n(n− 4)(n2 − 4)

16u

n+4n−4 . (4)

Just as in the scalar curvature setting, one can search for constant Q-curvature metrics in a conformal class by

minimizing the total Q-curvature. However, because of the conformal invariance one encounters the same lack of

compactness and presence of singular solutions.

In any event, a complete understanding of the fourth order analog of the Yamabe problem would require an

understanding of the following singular problem: let (M, g) be a compact Riemannian manifold and let Λ ⊂M be

55

56

a closed subset. A conformal metric g = u4

n−4 g is a singular constant Q-curvature metric if Qg is constant and g is

complete on M\Λ. According to (2) we can write this geometric problem as

Pgu =n(n− 4)(n2 − 4)

16u

n+4n−4 on M\Λ, (5)

lim infx→x0

u(x) = ∞ for each x0 ∈ Λ.

For the remainder of our work we concentrate on the case that (M, g) = (Sn,g) is the round metric on the sphere

and Λ = p1, . . . , pk is a finite set of distinct points. Thus we examine, given a singular set Λ with #(Λ) = k, the

set of functions

u : Sn\Λ = Sn\p1, . . . , pk → (0,∞)

that satisfy

Pu = P

gu =

n(n− 4)(n2 − 4)

16u

n+4n−4 (6)

lim infx→pj

u(x) = ∞ for each j = 1, 2, . . . , k.

For technical reasons we will also require Rg ≥ 0.

Following [1] we define the (unmarked) moduli space

Mk =

g ∈ [

g] : Qg =

n(n2 − 4)

8, Rg ≥ 0, g is complete on Sn\Λ, #(Λ) = k

. (7)

We equip each moduli space with the Gromov–Hausdorff topology. In the present work we explore some of the

structure of Mk when k ≥ 3. Let Λ = p1, . . . , pk with k ≥ 3 and let g = u4

n−4g ∈ MΛ. As it happens, the

metric g is asymptotic to a Delaunay metric near each puncture pj , and so one can associate a Delaunay parameter

ϵj(g) ∈ (0, ϵn] to each pj and g ∈ MΛ.

2 Main Results

Our main compactness theorem is the following.

Theorem 2.1. Let k ≥ 3 and let δ1 > 0, δ2 > 0 be positive numbers. Then the set

Ωδ1,δ2 = g ∈ Mk : distg(pj , pl) ≥ δ1 for each j = l, ϵj(g) ≥ δ2

is sequentially compact in the Gromov–Hausdorff topology.

References

[1] D. Pollack. Compactness results for complete metrics of constant positive scalar curvature on subdomains of

Sn. Indiana Univ. Math. J. 42 (1993), 1441–1456.


GEOMETRIC GRADIENT ESTIMATES FOR NONLINEAR PDES WITH UNBALANCED

DEGENERACY

JOAO VITOR DA SILVA1

1Departamento de Matematica - IMECC, UNICAMP, SP, Brasil, [email protected]

Abstract

We establish sharp C1,βloc geometric regularity estimates for bounded solutions of a class of nonlinear elliptic

equations with non-homogeneous degeneracy, whose model equation is given by

[|Du|p + a(x)|Du|q]∆u(x) = f(x) in Ω,

for a bounded and open set Ω ⊂ RN , and appropriate data p, q ∈ (0,∞), a and f . Such regularity estimates

simplify and generalize, to some extent, earlier ones via a different modus operandi. In the end, we present some

connections of our findings with a variety of relevant nonlinear models in the theory of elliptic PDEs.

1 Introduction

In this work we shall derive sharp C1.βloc geometric regularity estimates for solutions of a class of nonlinear elliptic

equations having a non-homogeneous double degeneracy, whose mathematical model is given by

[|Du|p + a(x)|Du|q] ∆u(x) = f(x) in Ω, (1)

for a β ∈ (0, 1), a bounded and open set Ω ⊂ RN and f ∈ C0(Ω) ∩ L∞(Ω).

In our studies, we enforce that the diffusion properties of the model (1) degenerate along an a priori unknown

set of singular points of solutions:

S0(u,Ω′) = x ∈ Ω′ ⋐ Ω : |Du(x)| = 0.

In turn, regarding the non-homogeneous degeneracy

Kp,q,a(x, |ξ|) = |ξ|p + a(x)|ξ|q, for (x, ξ) ∈ Ω × RN .

we shall assume that the exponents p, q and the modulating function a(·) fulfil

0 < p ≤ q <∞ and a ∈ C0(Ω, [0,∞)). (2)

Mathematically, (1) consists of a new model case of a nonlinear elliptic equation enjoying a non-homogeneous

degenerate term, which constitutes a non-divergent counterpart of certain variational integrals of the calculus of

variations with non-standard growth conditions as follows(W 1,p

0 (Ω) + u0, Lm(Ω)

)∋ (w, f) 7→ min

∫Ω

(Kp,q,a(x, |Dw|) − fw) dx, (DPF)

where a ∈ C0,α(Ω, [0,∞)), for some 0 < α ≤ 1, 1 < p ≤ q < ∞ and m ∈ (N,∞], see [2] and [5] for enlightening

works. Moreover, the Euler-Lagrange equation to (DPF) exhibits a type of non-uniform and doubly degenerate

ellipticity, which mixes up two different kinds of p−Laplacian type operators:

−div(A(x,∇u)) = f(x) with A(x, ξ) = p|ξ|p−2ξ + qa(x)|ξ|q−2ξ.

57

58

2 Main Results

We are in a position to state our main results.

Theorem 2.1 ([3, Theorem 1.1]). Assume that assumption (2) there hold. Let u be a bounded viscosity solution

to (1). Then, u is C1, 1p+1 , at interior points. More precisely, for any point x0 ∈ Ω′ ⋐ Ω there holds

[u]C

1, 1p+1 (Br(x0))

≤ C(universal) ·(∥u∥L∞(Ω) + 1 + ∥f∥

1p+1

L∞(Ω)

)for r ∈

(0,

1

2

).

An interpretation to Theorem 2.2 says that if u solves (1) and x0 ∈ Sr, 1p+1

(u,Ω′), then near x0 we obtain

supBr(x0)

|u(x)| ≤ |u(x0)| + C · r1+1

p+1 , where Sr, 1p+1

(u,Ω′) =x0 ∈ Ω′ ⋐ Ω : |Du(x0)| ≤ r

1p+1

.

On the other hand, from a geometric viewpoint, it is a pivotal qualitative information to obtain the (counterpart)

sharp lower bound estimate for such operators with non-homogeneous degeneracy.

Theorem 2.2 ([3, Theorem 1.2]). Suppose that the assumptions of Theorem 2.2 are in force. Let u be a

bounded viscosity solution to (1) with f(x) ≥ m > 0 in Ω. Given x0 ∈ Sr, 1p+1

(u,Ω′), there exists a constant

c = c(m, ∥a∥L∞(Ω), N, p, q,Ω) > 0, such that

sup∂Br(x0)

u(x) ≥ u(x0) + c · r1+1

p+1 for all r ∈(

0,1

2

).

Our findings extend/generalize regarding non-variational scenario, former results (Holder gradient estimates)

from [1, Theorem 3.1 and Corollary 3.2] and [1, Theorem 1], and to some extent, those from [3, Theorem 1]

by making using of different approaches and techniques adapted to the general framework of the nonlinear and

non-homogeneous degeneracy models.

References

[1] araujo, d.j., ricarte, g.c. and teixeira, e.v. - Geometric gradient estimates for solutions to degenerate

elliptic equations. Calc. Var. Partial Differential Equations 53 (2015), 605-625.

[2] colombo, m and mingione, g - Regularity for double phase variational problems. Arch. Rational Mech.

Anal. 215 (2015) 443-496.

[3] da silva, j.v. and ricarte, g. c. - Geometric gradient estimates for fully nonlinear models with non-

homogeneous degeneracy and applications. Calc. Var. Partial Differential Equations 59, 161 (2020).

[4] de filippis, c. - Regularity for solutions of fully nonlinear elliptic equations with nonhomogeneous degeneracy.

Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), no. 1, 110-132.

[5] de filippis, c. and mingione, g. - On the Regularity of Minima of Non-autonomous functionals. The Journal

of Geometric Analysis, 30 (2) (2020) 1584-1626.

[6] imbert, c. and silvestre, l. - C1,α regularity of solutions of degenerate fully non-linear elliptic equations.

Adv. Math. 233 (2013), 196-206.


UM PROBLEMA ANISOTROPICO ENVOLVENDO O OPERADOR 1-LAPLACIANO COM PESOS

ILIMITADOS

JUAN C. ORTIZ CHATA1, MARCOS T. DE OLIVEIRA PIMENTA2 & SERGIO S. DE LEON3

1Instituto de Biociencias, Letras e Ciencias Exatas, UNESP, SP, Brasil, [email protected],2Faculdade de Ciencias Extas e da Terra, UNESP, SP, Brasil, [email protected],

3Universidad de Valencia, UV, Espanha, [email protected]

Abstract

Neste trabalho provamos a existencia de solucoes de variacao limitada para problemas elıpticos quasilineares

envolvendo o operador 1-laplaciano com peso, os quais tem a pecularidade de que tanto o peso desse operador

quanto o da nao linearidade sao ilimitados. Assim e necessario a definicao de um espaco de funcoes de variacao

limitada com peso para tratar esse tipo de problemas. Alem disso, utilizando uma versao da bem conhecida

desigualdade de Caffarelli-Kohn-Nirenberg estabelece-se a compacidade das imersoes continuas e compactas desse

espaco em alguns espacos de Lebesgue com peso. Tambem estende-se a teoria de paridade de Anzellotti e usa-se

uma variante do Teorema do Passo da Montanha.

1 Introducao

A classica desigualdade de Caffarelli-Kohn-Nirenberg (ver [5]) da uma interpolacao entre as normas de Lebesgue

com peso de funcoes e suas derivadas, a qual, por sua vez, estabelece a compacidade das imersoes contınuas e

compactas para os espacos de Sobolev com peso. Sendo esses espacos aplicados para o analises de varios problemas

elıpticos envolvendo os operadores laplaciano e p-laplaciano com peso, esse ultimo para p > 1 (ver [1, 9, 4, 3, 6]).

No caso de problemas envolvendo o operador 1−laplaciano com peso, cujo peso seja uma funcao limitada que

esteja longe de zero, o espaco natural para analizar esse tipo de problemas e o espaco das funcoes de variacao

limitada BV (ver [8, 7, 10]).

O objetivo deste trabalho e lidar com problemas envolvendo o operador 1−laplaciano com pesos ilimitados, onde

tais pesos estao relacionados com os da desigualdade de Caffarelli-Kohn-Nirenberg. Mais precisamente, estudamos

a existencia de solucoes nao negativas para o seguinte problema −div

(1

|x|aDu

|Du|

)=

1

|x|bf(u) em Ω,

u = 0 sobre ∂Ω,(1)

onde Ω e um conjunto aberto e limitado em RN (N ≥ 2) contendo a origem e com fronteira Lipschitz ∂Ω, e os dois

parametros satisfazem: 0 < a < N − 1 e a 0 e 1 < q < NN−(1+a−b) , tais que

|f(s)| ≤ c1 + c2sq−1, s ∈ [0,+∞);

(f4) Existe µ > 1 e s0 > 0 tais que

0 < µF (s) ≤ f(s)s, ∀ s ≥ s0,

onde F (t) =∫ t

0f(s)ds;

59

60

(f5) f e crescente em [0,+∞).


Teorema 2.1. Suponha que f satisfaz as condicoes (f1)−(f4). Entao existe uma solucao nao trivial e nao negativa

para o Problema (1). Essa solucao e uma solucao de menor energia se assume-se tambem a condicao (f5).

Prova: Duas abordagens diferentes serao usadas para provar esse resultado. En cada caso uma adequada variante

do Teorema do Passo da Montanha (ver [2]) e aplicado. No primeiro deles, consideramos solucoes aproximadas de

problemas envolvendo o operador p−laplaciano e depois, fazemos p → 1+. No segundo, aplicamos os metodos

varicionais versaties para, alem de mostrar a existencia de solucao, mostrar que a solucao possui a menor energia

entre todas as demais.

References

[1] B. Abdellaoui, E. Colorado and I. Peral, Some remarks on elliptic equations with singular potentials

and mixed boundary conditions. Adv. Nonlinear Stud. 4, No. 4, 503-533 (2004).

[2] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications.

J. Funct. Anal. 14, 349-381, 1973.

[3] R.B. Assuncao, P.C. Carriao, O.H. Miyagaki, Subcritical perturbations of a singular quasilinear elliptic

equation involving the critical Hardy-Sobolev exponent. Nonlinear Anal., 66, 1351-1364, 2007.

[4] F. Brock, L. Iturriaga, J. Sanchez and P. Ubilla, Existence of positive solutions for p-Laplacian

problems with weights. Communications on Pure and Applied Analysis, 5, no. 4, 941, 2006.

[5] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights. Compositio

Mathematica, 53, 259 - 275, 1984.

[6] P.C. Carriao, D.G. de Figueiredo and O.H. Miyagaki, Quasilinear elliptic equations of the Henon-type:

existence of non-radial solutions. Commun. Contemp. Math., 11, no. 5, 783 - 798, 2009.

[7] J.M. Mazon, The Euler-Lagrange equation for the anisotropic least gradient problem, Nonlinear Anal. Real

World Appl., 31, 452 - 472, 2016.

[8] J.S. Moll, The anisotropic total variation flow., Math. Ann. 332, no. 1, 177 - 218, 2005.

[9] B. Xuan, The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights. Nonlinear Anal.,

Theory Methods Appl., Ser. A, Theory Methods 62, No. 4, 703-725, 2005.

[10] A. Zuniga, Continuity of minimizers to weighted least gradient problems, Nonlinear Anal., Theory Methods

Appl., Ser. A, Theory Methods 178, 86–109, 2019.


COMPACT EMBEDDING THEOREMS AND A LIONS’ TYPE LEMMA FOR FRACTIONAL

ORLICZ–SOBOLEV SPACES

MARCOS L. M. CARVALHO1, EDCARLOS D. SILVA2, J. C. DE ALBUQUERQUE3 & S. BAHROUNI4

1Instituto de Matematica, UFG, GO, Brasil, marcos leandro [email protected],2Instituto de Matematica, UFG, GO, Brasil, [email protected],

3Departamento de Matematica, UFPE, PE, Brasil, [email protected],4Mathematics Department, University of Monastir, Tunisia, [email protected]

Abstract

In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces.

Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz

spaces, provided the weight is unbounded. We also obtain a version of Lions’ “vanishing” Lemma for fractional

Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law.

Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the

existence of ground state solutions for a class of nonlinear Schrodinger equations, taking into account unbounded

or bounded potentials.

1 Introduction

This work is motivated by a very recent trend in the fractional framework, which is to consider a new nonlocal and

nonlinear operator, the so-called fractional Φ-Laplacian. Throughout this work, we shall consider Φ : R → R an

even function defined by

Φ(t) =

∫ t

0

sφ(s) ds,

where φ : R → R is a C1-function satisfying the following assumptions:

(φ1) tφ(t) is strictly increasing in (0,∞) such that tφ(t) 7→ 0, as t 7→ 0 and tφ(t) 7→ ∞, as t 7→ ∞;

(φ3) there exist ℓ,m ∈ (1, N) such that ℓ ≤ t2φ(t)Φ(t) ≤ m < ℓ∗, for all t > 0.

For s ∈ (0, 1) and u smooth enough, the fractional Φ-Laplacian operator is defined as

(−∆Φ)su(x) := P.V.

∫φ (|Dsu|)

Dsu

|x− y|N+sdy, where Dsu :=

u(x) − u(y)

|x− y|s(1)

and P.V. denotes the principal value of the integral. Note that if φ(t) = tp−2, p ∈ (1, N) then (1) reduces to the

fractional p-Laplace operator. In a similar way, if φ(t) = tp−2 + tq−2, 1 < p < q < N , then we have the fractional

(p, q)-Laplacian operator.

Due to the generality of the fractional Φ-Laplacian operator (1) and motivated by the very recent papers,

mainly taking into account the work of Bonder and Salort [1], our goal is to study the following class of fractional

Schrodinger equations

(−∆Φ)su+ V (x)φ(u)u = f(x, u), x ∈ RN , (P )

where N > 2s, 0 < s < 1. The potential satisfies the following assumptions:

(V0) It holds that V (x) ≥ V0 for any x ∈ RN where V0 > 0;

61

62

(V1) The set x ∈ RN ;V (x) < M has finite Lebesque measure for each M > 0.

The nonlinear term f is of C1 class and satisfies suitable assumptions.

Due to the presence of the potential V (x), we introduce the following suitable weighted fractional Orlicz-Sobolev

space

X :=

u ∈W s,Φ(RN ) :

∫V (x)Φ(|u|) dx < +∞

,

endowed with the norm

∥u∥ = [u]s,Φ + ∥u∥V,Φ,

where

∥u∥V,Φ = inf

λ > 0 :

∫RN

V (x)Φ

(u(x)

λ

)dx ≤ 1

and the (s,Φ)-Gagliardo semi-norm is defined as

[u]s,Φ := inf

λ > 0:

∫ ∫RN×RN

Φ

(u(x) − u(y)

λ|x− y|s

)dxdy

|x− y|N≤ 1

.

2 Main Results

Our main contribution

Theorem 2.1 (Compact embedding). Assume that (φ1) − (φ2) and (V0) − (V1) hold. Then, the embedding

X → LΦ(RN ) is compact.

Theorem 2.2 (Compact embedding). Assume that (φ1)–(φ2) and (V0) − (V1) hold. Suppose that Φ ≺ Ψ ≺≺ Φ∗

and the following limit holds

lim sup|t|→0

Ψ(|t|)Φ(|t|)

< +∞. (1)

Then, the space X is compactly embedded into LΨ(RN ).

Theorem 2.3 (Lions’ Lemma type result). Suppose that (φ1) − (φ2) hold and

lim|t|→0

Ψ(t)

Φ(t)= 0. (2)

Let (un) be a bounded sequence in W s,Φ(RN ) in such way that un 0 in X and

limn→+∞

[supy∈RN

∫Br(y)

Φ(un) dx

]= 0, (3)

for some r > 0. Then, un → 0 in LΨ(RN ), where Ψ is an N -function such that Ψ ≺≺ Φ∗.

To prove the above results, we shall introduce new techniques to overcome the lack of a suitable interpolation

law. Finally, we shall apply these results to obtain solutions to the Problem (P ).

References

[1] J. F. Bonder and A. M. Salort, Fractional order Orlicz-Sobolev spaces, Journal of Functional Analysis, 277

(2019), 333-367.

[2] Carvalho, M. L., Silva, E., Albuquerque, J. C. r.h. and Bahrouni, S. -Compact embedding theorems

and a Lions’ type Lemma for fractional Orlicz–Sobolev spaces,arxiv.org/abs/2010.10277v1, (2020).


COUPLED AND UNCOUPLED SIGN-CHANGING SPIKES OF SINGULARLY PERTURBED

ELLIPTIC SYSTEMS

MAYRA SOARES1 & MONICA CLAPP2

1Instituto de Matematicas, UNAM, CDMX, Mexico, ssc [email protected],2Instituto de Matematicas, UNAM, CDMX, Mexico, [email protected]

Abstract

We study the singularly perturbed system of elliptic equations−ϵ2∆ui + ui = µi|ui|p−2ui +

ℓ∑j=1j =i

λijβij |uj |αij |ui|βij−2ui,

ui ∈ H10 (Ω), ui = 0, i = 1, . . . , ℓ,

(1)

in a bounded domain Ω in RN , with N ≥ 3, ϵ > 0, µi > 0, λij = λji < 0, αij , βij > 1, αij = βji,

αij+βij = p ∈ (2, 2∗), and 2∗ := 2NN−2

. If Ω is the unit ball, we obtain solutions with a prescribed combination of

positive and nonradial sign-changing components exhibiting two different types of asymptotic behavior as ϵ→ 0:

solutions whose limit profile is a rescaling of a solution with positive and nonradial sign-changing components

of the limit system −∆ui + ui = µi|ui|p−2ui +

ℓ∑j=1j =i

λijβij |uj |αij |ui|βij−2ui,

ui ∈ H1(RN ), ui = 0, i = 1, . . . , ℓ,

(2)

and solutions whose limit profile is a solution of the uncoupled system, i.e., after rescaling and translation, the

limit profile of the i-th component is a positive or a nonradial sign-changing solution to the to the problem

−∆u+ u = µi|u|p−2u, u ∈ H1(RN ), u = 0. (3)

1 Introduction

System (1) arises as a model for various physical phenomena, in particular in the study of standing waves for

a mixture of Bose-Einstein condensates of ℓ different hyperfine states which overlap in space, see for example [1].

Here we consider the case in which the interaction between particles in the same state is attractive (µi > 0) and

the interaction between particles in any two different states is repulsive (λij < 0). Our main objective is to study

the existence and profile of solutions to (1) some of whose components can be positive while others change sign.

It is reasonable to expect that there will be solutions with sign-changing spikes, i.e., solutions whose sign-changing

components look like rescaling of a sign-changing solution of (3). On the other hand, rescaling the components by

ui(x) := ui(ϵx) system (1) becomes system (2) in Ωϵ := x ∈ RN : ϵx ∈ Ω instead of RN . As ϵ→ 0 these domains

cover the whole space RN . So it is natural to ask if the system (1) has a solution that, after rescaling, approaches a

solution to the system (2). One might also expect to obtain solutions with positive and sign-changing components

for the system (1) whose limit profile is a solution of the same type for the system (2).

2 Main Results

Our main result are read as follows:

63

64

Theorem 2.1. Let N = 4 or N ≥ 6. Then, for any given 0 ≤ m ≤ ℓ, the system (2) has a solution

w = (w1, . . . ,wℓ) whose first m components w1, . . . , wm are positive and whose last ℓ−m components wm+1, . . . , wℓ

are nonradial and change sign. Furthermore, w satisfieswi(z1, z2, x) = wi(eiϑz1, e

iϑz2, gx) for all ϑ ∈ [0, 2π), g ∈ O(N − 4), i = 1, . . . , ℓ,

wi(z1, z2, x) = wi(z2, z1, x)if i = 1, . . . ,m, wi(z1, z2, x) = −wi(z2, z1, x)if i = m+ 1, . . . , ℓ,(1)

for all (z1, z2, x) ∈ C×C×RN−4 ≡ RN , and it has least energy among all nontrivial solutions with these properties.

To illustrate our results, let us focus on the case where Ω is the open unit ball B1(0) in RN centered at the

origin. For ϵ > 0 and u ∈ H1(RN ) let ∥u∥2ϵ :=1

ϵN

∫RN

[ϵ2|∇u|2 + u2

]and ∥u∥ := ∥u∥1.

Theorem 2.2. Let N = 4 or N ≥ 6, and Ω = B1(0). Then, for any given 0 ≤ m ≤ ℓ and any sequence (ϵk)

of positive numbers converging to zero, there exists solution uk = (u1k, . . . , uℓk) to the system (3) whose first m

components are positive and whose last ℓ −m components are nonradial and change sign, with the following limit

profile: There exists a solution w = (w1, . . . ,wℓ) to the system (2) such that, after passing to a subsequence,

limk→∞

∥uik − wi(ϵ−1k · )∥ϵk = 0 for all i = 1, . . . , ℓ.

The first m components of w are positive, its last ℓ−m components are nonradial and change sign, and w satisfies

(1). Therefore, limk→∞

ℓ∑i=1

∥uik∥2ϵk =

ℓ∑i=1

∥wi∥2 =: cm.

Theorem 2.3. Let N ≥ 5 and Ω = B1(0). Then, for any given 0 ≤ m ≤ ℓ and any sequence (ϵk) of positive

numbers converging to zero, there exists solution uk = (u1k, . . . ,uℓk) to the system (1) with ϵ = ϵk and Ω = B1(0),

whose first m components are positive and whose last ℓ −m components are nonradial and change sign, with the

following limit profile: For each i = 1, . . . , ℓ, there exist a sequence (ξik) in B1(0) and a solution vi to the problem

(3) such that, after passing to a subsequence, limk→∞

ϵ−1k dist(ξik, ∂B1(0)) = ∞, lim

k→∞ϵ−1k |ξik − ξjk| = ∞ if i =

j, limk→∞

∥uik − vi(ϵ−1k ( · − ξik))∥ϵk = 0. The functions v1, . . . , vm are positive and radial, while the functions

vm+1, . . . , vℓ are sign-changing, nonradial and satisfyvi(z1, z2, x) = vi(eiϑz1, e

iϑz2, gx) for all ϑ ∈ [0, 2π), g ∈ O(N − 4),

vi(z1, z2, x) = −vi(z2, z1, x),(2)

for all (z1, z2, x) ∈ C×C×RN−4 ≡ RN , i = m+1, . . . , ℓ. Furthermore, limk→∞

ℓ∑i=1

∥uik∥2ϵk =

ℓ∑i=1

∥vi∥2 =: cm, satisfies

cm < cm, with cm as in Theorem 2.2, if N ≥ 6.

References

[1] Bohn, J. L.; Burke Jr., J. P.; Esry, B. D.; Greene, C. H.: Hartree-Fock theory for double condensates, Phys.

Rev. Lett., 78, 3594-3597 (1997).

[2] Clapp, M.; Srikanth, P. N.: Entire nodal solutions of a semilinear elliptic equation and their effect on

concentration phenomena, J. Math.Anal.Appl. 437, 485-497 (2016).

[3] Clapp, M.; Szulkin, Andrzej: A simple variational approach to weakly coupled competitive elliptic systems.

NoDEA Nonlinear Differential Equations Appl., 26 no. 4, Paper No. 26, (2019).


ON AN AMBROSETTI-PRODI TYPE PROBLEM IN RN

CLAUDIANOR O. ALVES1, ROMILDO N. DE LIMA2 & ALANNIO B. NOBREGA3

1Unidade Academica de Matematica, UFCG, PB, Brasil, [email protected],2Unidade Academica de Matematica, UFCG, PB, Brasil, [email protected],3Unidade Academica de Matematica, UFCG, PB, Brasil, [email protected]

Abstract

In this work we study results of existence and non-existence of solutions for the following Ambrosetti-Prodi

type problem −∆u = P (x)

(g(u) + f(x)

)in RN ,

u ∈ D1,2(RN ), lim|x|→+∞ u(x) = 0,(P )

where N ≥ 3, P ∈ C(RN ,R+), f ∈ C(RN ) ∩ L∞(RN ) and g ∈ C1(R). The main tools used are the sub-

supersolution method and Leray-Schauder topological degree theory.

1 Introduction

The main motivation to study the problem (P ) comes from the seminal paper by Ambrosetti and Prodi [2] that

studied the existence and non-existence of solution for the problem−∆u = g(u) + f(x), in Ω,

u = 0, in ∂Ω,(1)

where Ω ⊂ RN with N ≥ 3, is a bounded domain, g is a C2−function with

g′′(s) > 0, ∀s ∈ R and 0 < lims→−∞

g′(s) < λ1 < lims→∞

g′(s) < λ2.

In order to prove their results, Ambrosetti and Prodi used a global result of inversion to proper functions to show

the existence of a closed manifold M dividing the space C0,α(Ω) in two connected components O1 and O2 such

that:

(i) If f belongs to O1, the problem (1) has no solution;

(ii) If f belongs to M , the problem (1) has exactly one solution;

(iii) If f belongs to O2, the problem (1) has exactly two solution.

In [3], Berger and Podolak proposed the decomposition of function f in the form f = tϕ + f1, where ϕ is

eigenfunction associated to first eigenvalue of ”−∆”−∆u = g(u) + tϕ+ f1, in Ω,

u = 0, in ∂Ω,(2)

then using the Liapunov-Schmidt method they showed the existence of t0 ∈ R such that (2) has at least two

solutions if t < t0, at least one solution if t = t0 and no solutions if t > t0.

Now, before stating our main results, we need to fix the assumptions on the functions P and g. In the sequel,

g : R → R is a C1−function that satisfies the following inequalities

lim sups→−∞

g(s)

s< λ1 < lim inf

s→∞

g(s)

s(G1)

65

66

Related to the function P : RN → R+, we consider that it is a continuous function satisfying:

| · |2P (·) ∈ L1(RN ) ∩ L∞(RN ) (P1)

∫RN

P (y)

|x− y|N−2dy ≤ C

|x|N−2,∀x ∈ RN \ 0, for some C > 0 (P2)

We denote by N the eigenspace associated with the first eigenvalue λ1. By [1], it is well known that

dimN = 1, then we can assume that N = Spanϕ, where ϕ is one positive eigenfunction associated with λ1

with∫RN P (x)|ϕ|2dx = 1. Hence, we can write f = tϕ+ f1, where f1 ∈ C(RN ) ∩ L∞(RN ) with∫

RN

P (x)f1ϕdx = 0 and

∫RN

P (x)fϕdx = t. (3)

From this, problem (P ) can be rewritten as follows−∆u = P (x)

(g(u) + tϕ(x) + f1(x)

)in RN ,

u ∈ D1,2(RN ), lim|x|→+∞ u(x) = 0.(P )

2 Main Results

Our first result is the following:

Theorem 2.1. Assume the conditions (G1), (P1) and (P2). Then, for each f1 ∈ N⊥ there is a number α(f1) such

that:

(i) The problem (P ) has no solution whenever t > α(f1);

(ii) If t < α(f1), then (P ) has at least one solution.

Our second result is the following:

Theorem 2.2. Assume the conditions (G1), (P1) and (P2). Moreover, assume that g is an increasing function

satisfying

lims→+∞

g(s)

sσ= 0, (1)

where σ = NN−2 . Then, for each f1 ∈ N⊥ there is a number α(f1) such that:

(i) If t < α(f1), then (P ) has at least two solutions;

(ii) If t = α(f1), then (P ) has at least one solution.

References

[1] C. O. Alves, R. N. de Lima and M. A. S. Souto, Existence of a solution for a non-local problem in RN via

bifurcation theory, Proc. Edin. Math. Soc., 61 , 825-845 (2018).

[2] A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between

Banach spaces, Ann. Mat. Pura Appl. 93 (1972), 231-246.

[3] M.S. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J. 24

(1974/1975) 837-846.


AN ELLIPTIC SYSTEM WITH MEASURABLE COEFFICIENTS AND SINGULAR

NONLINEARITIES

LUCIO BOCCARDO1, STEFANO BUCCHERI2 & CARLOS ALBERTO PEREIRA DOS SANTOS3

1Sapienza UniversitA di Roma, Italia, [email protected],2University of Vienna, Austria, [email protected],3Universidade de Brasılia, DF, Brasil, [email protected]

Abstract

In this talk we present some new existence results for a system of elliptic equations with a singular nonlinearity.

Our approach is based on a comparison principle for weak solutions and the Schauder fixed point Theorem. The

difficulties due to the presence of the singularity are tackled using suitable test functions and barriers. Despite

our strategy is not variational, as a byproduct of our results, we are able to find saddle points of the functional

associated to the system.

1 Introduction

In this talk we will focus on the following system with a singular nonlinearity

−div(A(x)u) + vur−1 =1

uγin Ω ,

−div(M(x)v) = ur in Ω ,

u > 0 in Ω ,

u = φ = 0 on ∂Ω ,

(1)

where Ω is bounded open subset of RN ; the two exponents satisfy γ > 0 and r > max0, 1− γ and the measurable

matrices A(x),M(x) are elliptic in the sense that

α|ξ|2 ≤ A(x) ξ ξ ≤ β|ξ|2 and α|ξ|2 ≤M(x) ξ ξ ≤ β|ξ|2 , (2)

for almost every x in Ω, and for every ξ in RN , with 0 < α ≤ β.

We stress that when γ ∈ (0, 1) even the nonlinear term in the left hand side of the first equation in (1) can be

singular. Anyway this singularity mild compared with the one on the right hand side (see assumption on r).

The literature about singular equation is wide and well establish. Without the intention of being exhaustive we

mention the seminal papers [1], [2] and [3]. We stress that in the previously mentioned paper the singularity is of

reaction type, meaning something like

−div(A(x)u) = h(u) with h(u) ≈ 1

|u|γnear the origin.

If the singularity is of the absorption type, namely something like

−div(A(x)u) + h(u) = 1 with h(u) ≈ 1

|u|γnear the origin,

the features of the problem change dramatically. As we already pointed out, since r is always greater then 1 − γ,

in broad terms our problem is part of the first setting.

67

68

2 Main Results

For the sake of brevity here we present the result concerning solutions in the energy space and γ ≤ 1. For the more

general case we refer to the manuscript [3].

Definition 2.1. A couple of functions (u, v) ∈(W 1,2

0 (Ω) ∩ L(Ω)∞)2

is a energy solution to system (1) if

u, v > 0 a.e. in Ω,

ϕ

uγ∈ L(Ω)1 ∀ ϕ ∈W 1,2

0 (Ω)

and if

∫Ω

A(x)∇u∇ϕ+

∫Ω

vur−1ϕ =

∫Ω

ϕ

uγ∀ ϕ ∈W 1,2

0 (Ω)

∫Ω

M(x)∇v∇ψ =

∫Ω

urψ ∀ ψ ∈W 1,20 (Ω).

(1)

Theorem 2.1. Let Ω be a bounded open set of RN and assume (2). Given γ ∈ (0, 1] and r > 1 − γ, there exists

(u, v) ∈(W 1,2

0 (Ω) ∩ L(Ω)∞)2

energy solution to (1). Moreover such a couple is a saddle point to the functional

J(w, z) =1

2

∫Ω

A(x)∇w∇w − 1

2r

∫Ω

M(x)∇z∇z +1

r

∫Ω

z+|w|r − 1

1 − γ

∫Ω

(w+)1−γ

,

namely

J(u, z) ≤ J(u, v) ≤ J(w, v) for any (w, z) ∈(W 1,2

0 (Ω))2.

References

[1] M.G. Crandall, P.H. Rabinowitz, L. Tartar, On a Dirichletproblem with a singular nonlinearity, Comm. Partial

Differential Equations 2(2) (1977) 193-222.

[2] A.C. Lazer, P.J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc.

111(3) (1991) 721-730

[3] L. Boccardo, L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential

Equations 37(3-4) (2010) 363-380.

[4] L. Boccardo, S. Buccheri, C. A. Santos, An elliptic system with measurable coefficients and singular

nonlinearities, manuscript.


FOURTH-ORDER NONLOCAL TYPE ELLIPTIC PROBLEMS WITH INDEFINITE

NONLINEARITIES

EDCARLOS D. DA SILVA1, THIAGO R. CAVALCANTE2 & J.C. DE ALBUQUERQUE3

1Instituto de Matematica, UFG, GO, Brasil, [email protected],2Departamento de Matematica, UFT, TO, Brasil, [email protected],

3Departamento de Matematica, UFPE, PE, Brasil, [email protected]

Abstract

In this work we establish the existence of at least one weak solution and one ground state solution for the

following class of fourth-order nonlocal elliptic problems∆2u− g

(∫Ω

|∇u|2 dx)∆u = µa(x)|u|q−2u+ b(x)|u|p−2u in Ω,

u = ∆u = 0 on ∂Ω,

where N ≥ 5, Ω ⊂ RN is a smooth bounded domain, ∆2 = ∆ ∆ is the biharmonic operator, µ > 0,

1 < q < 2 < p < 2N/(N −4) and g : [0,∞) → [0,∞) satisfies suitable assumptions. We deal with the case where

a, b : Ω → R can be sign changing functions, which means that the problem is indefinite. Our approach is based

on variational methods jointly with a fine analysis on the Nehari manifold, by giving a complete description of

the fibering maps, which strongly depend on the sign of the weights.

1 Introduction

In this work we study the following class of fourth-order elliptic problems∆2u− g

(∫Ω

|∇u|2 dx

)∆u = µa(x)|u|q−2u+ b(x)|u|p−2u in Ω,

u = ∆u = 0 on ∂Ω,

(Pµ)

where ∆2 = ∆ ∆ is the biharmonic operator, µ > 0, N ≥ 5, Ω ⊂ RN is a smooth bounded domain and

1 < q < 2 < p < 2∗, where 2∗ := 2N/(N − 4) is the critical Sobolev exponent. The function g is a smooth function

satisfying some assumptions and a, b can be sign changing functions. Before introducing our assumptions and main

results, we give a brief survey on the related results, which motivate this work.

2 Main Results

Throughout this work we suppose that a, b : Ω → R are bounded which can be sign changing functions and b satisfies

the following condition:

(A) There exist Ω0,Ω1 ⊂ Ω with |Ω0|, |Ω1| > 0, such that a(x) > 0 for all x ∈ Ω0 and b(x) > 0, for all x ∈ Ω1,

where | · | denotes the Lebesgue measure.

For the function g ∈ C2([0,+∞), [0,+∞)) we shall consider the following conditions:

(G1) The function g is nonnegative and nondecreasing.

69

70

(G2) There exists r ≥ 2/(p− 2) such that

g(t) ≥ rg′(t)t, for all t ≥ 0.

(G3) There exist σ ∈ (2/p, 1) and m ∈ (2/p, 2/q) such that

σg(t)t ≤ G(t) ≤ mg(t)t, for all t ≥ 0,

where G(t) =∫ t

0g(s) ds, t ∈ R.

(G4) There exists ρ ∈ (2/(p− 4),∞) such that

g′(t) ≥ ρg′′(t)t, for all t ≥ 0.

(G5) There exist constants c1, c2 > 0 and k < (p− 2)/2 such that

g(t) ≤ c1 + c2tk, for all t ≥ 0.

The first main result of this paper can be stated as follows:

Theorem 2.1. Suppose that 1 < q < 2 0 such that Problem (Pµ) has at least two nontrivial solutions u1, u2 ∈ H satisfying Jµ(u1) < 0 < Jµ(u2),

whenever µ ∈ (0, µ⋆). Furthermore, u1 is a ground state solution, that is, u1 has the least energy level among all

nontrivial solutions of (Pµ).

References

[1] K. J. Brown, T. F. Wu A Fibering map approach to a semilinear elliptic boundary value problem, Electronic

Journal of Differential Equations, 69, (2007), 1-9.

[2] D.G. de Figueiredo, J.P. Gossez, P. Ubilla Local superlinearity and sublinearity for indefinite semilinear

elliptic problems, J. Funct. Anal. 199 (2003), no. 2, 452–467.

[3] Giovany M. Figueiredo, Marcelo F. Furtado, Joao Pablo P. da Silva, Two solutions for a fourth

order nonlocal problem with indefinite potentials, Manuscripta math. 160, (2019), 199–215.

[4] M. Furtado, E. da Silva, Superlinear elliptic problems under the Nonquadriticty condition at infinity, Proc.

Roy. Soc. Edinburgh Sect. A 145 (2015), no. 4, 779–790.

[5] G. Kirchhoff, Mechanik, Teubner, Leipzig, (1883).

[6] T.R. Cavalcante, E. da Silva Multiplicity of solutions to fourth-order superlinear elliptic problems under

Navier conditions, Electron. J. Differential Equations 167 (2017), 1-16.

[7] Giovany M. Figueiredo, Joao R. Junior Santos, Multiplicity of solutions for a Kirchhoff equation with

subcritical or critical growth, Differential Integral Equations 25 (2012), no. 9-10, 853–868.

[8] F. Wang, M. Avci M., Y. An, Existence of solutions for fourth order elliptic equations of Kirchhoff type, J.

Math. Anal. Appl. 409(1), 140–146 (2014)


ON THE FRICTIONAL CONTACT PROBLEM OF P (X)-KIRCHHOFF TYPE

WILLY BARAHONA M.1, EUGENIO CABANILLAS L.2, ROCIO DE LA CRUZ M.3, JESUS LUQUE R.4 & HERON

MORALES M.5

1Instituto de Investigacion, FCM,UNMSM, Peru, [email protected],2Instituto de Investigacion, FCM,UNMSM, Peru, [email protected],3Instituto de Investigacion, FCM,UNMSM, Peru, [email protected],

4Instituto de Investigacion, FCM,UNMSM, Peru, [email protected],5Facultad de Ciencias, UNS, Peru, [email protected]

Abstract

In this article we consider a class of frictional contact problem of p(x)-Kirschhoff type. By means of an

abstract Lagrange multiplier technique and the Schauder’s fixed point theorem we establish the existence of

weak solutions.

1 Introduction

The purpose of this work is to investigate the existence of weak solutions for the following boundary value problem

−M(∫

Ω

1

p(x)|∇u|p(x)dx

)div(|∇u|p(x)−2∇u

)= f1(x, u) in Ω

u = 0 on Γ1

M(∫

Ω

1

p(x)|∇u|p(x)dx

)|∇u|p(x)−2 ∂u

∂ν= f2 in Γ2∣∣∣M(∫

Ω

1

p(x)|∇u|p(x)dx

)|∇u|p(x)−2 ∂u

∂ν

∣∣∣ ≤ g,

M(∫

Ω

1

p(x)|∇u|p(x)dx

)|∇u|p(x)−2 ∂u

∂ν= −g u

|u|, if u = 0 in Γ3

(1)

where Ω ⊆ R2 is a bounded domain with smooth enongh boundary Γ, partitioned in three parts Γ1,Γ2,Γ3 such

that meas (Γi) > 0, (i = 1, 2, 3); f1 : Ω × R → R, f2 : Γ2 → R, g : Γ3 → R and M : R+ → R+ are given functions,

p ∈ C(Ω).

The study of the p(x)-Kirchhoff type equations with nonlinear boundary condition of different class have attracted

expensive interest in recent years (See e.g. [1,3]). Motivated by the ideas in [2] we consider problem (1.1) (which

has already been treated for constant exponent, with M(s) = 1, f1(x, u) = f1(x)) with M a nonconstant continuous

function in the setting of the variable exponent spaces.

2 Assumptions and Main Result

First, we introduce the space

X = u ∈W 1,p(x)(Ω) : γu = 0 on Γ1

herein W 1,p(x)(Ω) (p ∈ C(Ω), 2 ≤ p(x) < +∞) is the well known variable exponent Sobolev space.

(A1) M : [0,+∞[→ [m0,+∞[ is a continuous and decreasing function; m0 > 0.

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(A2) f1 : Ω × R → is a Caratheodory function satisfying

|f1(x, t)| ≤ c1 + c2|t|α(x)−1 , ∀(x, t) ∈ Ω × R , α ∈ C+(Ω) , α(x) < p∗(x)

(A3) f2 ∈ Lp′(x)(Γ2), g ∈ Lp′(x)(Γ3), g(x) ≥ 0 a.e on Γ3

Now we introduce the spaces

S = u ∈W1− 1

p′(x),p(x)

(Γ) : ∃v ∈ X such that u = γv on Γ,

Y = S′ (the dual of the space S) and the set of Lagrange multipliers

Λ = u ∈ Y :⟨µ, z⟩≤∫Γ3

g(x)|z(x)|dΓ , ∀z ∈ S.

Next we define a Lagrange multiplier λ ∈ Y⟨µ, z⟩

= −∫Γ3

M( 1

p(x)|∇u|p(x)

)|∇u|p(x)−2 ∂u

∂νzdΓ , ∀z ∈ S.

So, our main result can be stated as follows.

Theorem 2.1. Suppose (A1) - (A3) hold. Then problem (1.1) has a solution (u, λ) ∈ X × Λ.

Proof We apply an abstract result on [4] and the Schauder’s fixed point theorem.

References

[1] Cammaroto F., Vilasi L. - Existence of three solutions for a Degenerate Kircchoff - type transmission

problem., Numer Funct Anal Optim, 35 (2014) 911-931.

[2] Cojocara M., Matei A. - Well - posedness for a class of frictional contact models via mixed variational

formulations, Nonlinear Anal RWA, 47(2019) 127-141.

[3] Handami M. K, Chang N. T. , Repovs D. D. - New class of Sixth - order nonhomogeneous p(x)-Kirchhoff

problems with sign - changing weight functions , Adv Nonlinear Anal, 10(1)(2021) 1117-1131.

[4] Matei A. - An existence result for a mixed variational problem arising from contact mechanics , Nonlinear

Anal RWA, 20(2014) 74-81.


GLOBAL MULTIPLICITY OF SOLUTIONS FOR A MODIFIED ELLIPTIC PROBLEM WITH

SINGULAR TERMS

JIAZHENG ZHOU1, CARLOS ALBERTO P. DOS SANTOS2 & MINBO YANG3

1Departamento de Matematica, UnB, DF, Brasil, [email protected],2Departamento de Matematica, UnB, DF, Brasil, [email protected],3Zhejiang Normal University, Jinghua, China, [email protected]

Abstract

We establish global multiplicity results of solutions for a singular nonlinear problem. We first prove a

comparison principle to prove the existence of a minimal solution by the method of sub and super solutions and

then we also obtain the second solution by critical point theory.

1 Introduction

In this paper, we are interested in the global multiplicity results and qualitative properties of the solutions for the

modified quasilinear equation with singular nonlinearities−∆u− u∆u2 = λa(x)u−α + b(x)uβ in Ω,

u > 0 in Ω, u = 0 on ∂Ω,(1)

where Ω ⊂ RN is a smooth bounded domain with N ≥ 3, 0 < a ∈ C(Ω) ∩ L∞(Ω), b ∈ C(Ω) ∩ L∞(Ω) with

b−/a ∈ L∞(Ω) 0 < α < 1 < β ≤ 22∗ − 1 and λ > 0 is a real parameter.

The study of the modified quasilinear Schrodinger equation in the whole space had received a lot of attention

in the last decades. The existence of a positive ground state solution of equation (1) has been proved in [1] by

introducing parameter λ in front of the nonlinear term. In [2], by changing of variables, the authors studied the

quasilinear problem was transformed to a semilinear one and the existence of a positive solution was proved by

the Mountain-Pass lemma in an Orlicz working space. Different from the changing variable methods, in [3] the

authors introduced a new perturbation techniques to study a class of subcritical quasilinear problems including the

Modified Schrodinger equation (1).

However, to the best of our knowledge, the problem of global multiplicity of solutions for the modified quasilinear

equation on bounded domain with sub critical and critical growth is still open. In the present paper we are going

to study the existence and global multiplicity of the solutions for the singular modified elliptic equation driven

by operator −∆u − u∆u2 and the critical term. We will observe that how the parameter λ and the nonconstant

functions a(x), b(x) will affect the existence and multiplicity of the solutions.

2 Main Results

Theorem 2.1 (Singular/superlinear-Global existence). Assume that a > 0, b+ = 0, and 0 < α < 1 < β ≤ 22∗ − 1.

Then there exists a 0 < λ⋆ < ∞ such that problem (1) admits at least one solution uλ ∈ H10 (Ω) for 0 < λ ≤ λ⋆,

and no H10 (Ω)-solution for λ > λ⋆. Besides this, we have:

(1) any solution of problem (1) belongs to L∞(Ω),

(2) that there exists τ0 > 0 such that uλ is the unique solution of the problem (1) with L∞(Ω)-norm smaller or

equal than τ0,

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(3) ||uλ|| = o(λθ) as λ→ 0, for any 0 < θ < 1/(1 + α),

(4) uλ is a minimal solution if b− ≡ 0 holds. In particular, ∥uλ∥ < ∥uµ∥ if 0 < λ < µ ≤ λ⋆ holds.

Proof The first, we will apply a suitabla change of variable to transform the quasilinear equation (1) into a

semilinear equation which remains singular at zero and behaves superlinearly at infinity:−∆ω = [λa(x)h(ω)−α + b(x)h(ω)β ]h′(ω) in Ω,

ω > 0 in Ω, ω = 0 on ∂Ω,(1)

where h : R → R satisfies h′(t) = (1 + 2|h(t)|2)−1/2, t > 0 and h(−t) = −h(t) for t ≤ 0.

The second, we will use the method of sub and super solutions and minimization to prove that there exists a

0 < λ∗ <∞ such that problem (1) admits at least one solution uλ = h(ωλ) ∈ H10 (Ω) for 0 < λ ≤ λ∗ and no solution

for λ > λ∗. Besides this, uλ satisfies (1), (2), (3) and (4) listed above.

Theorem 2.2 (Singular/superlinear-Global multiplicity). Assume that a > 0, b+ = 0, 0 < α < 1 and one of the

following assumptions:

(i) 1 < β < 22∗ − 1, (ii) b(x) ≡ 1 and β = 22∗ − 1.

Then the problem (1) admits a second solution vλ in H10 (Ω) ∩ L∞(Ω), with uλ ≤ vλ, for any 0 < λ < λ⋆ given.

Proof The second solution of (1) will be found by making the use of Ekeland’s variational Principle on the followin

set

T = ω ∈ H10 (Ω), ω ≥ ωλ a., e. in Ω

where ωλ is the first solution of (1). By the proof of Theorem 2.1, there exists 0 < l0 ≤ ||ωλ|| such that

Iλ(ω) ≥ Iλ(ωλ), ∀ω with ||ω − ωλ|| ≤ l0. Then one of the following cases holds:

(P1) infIλ(ω), ω ∈ T, ||ω − ωλ|| = l = Iλ(ωλ), ∀l ∈ (0, l0);

(P2) There exists l1 ∈ (0, l0) such that infIλ(ω), ω ∈ T, ||ω − ωλ|| = l1 > Iλ(ωλ),

where Iλ(ω) is the energy functional associated to (1).

If (P1) is true, we prove that there exists a solution ηλ of (1) such that ηλ ≤ ωλ in Ω and ||ωλ − ηλ|| = l for any

l ∈ (0, l0) and each λ ∈ (0, λ∗).

If (P2) is true, we prove that there exists a solution ηλ of (1) such that ηλ ≤ ωλ in Ω and ||ωλ − ηλ|| = l1 for

each λ ∈ (0, λ∗).

Finally, we take vλ = h(η) and we have that vλ = uλ is a second solution for (1).

References

[1] J. Liu and Z. Q. Wang, Solitonsolutions for quasilinear Schrodinger equations I, Proc. Amer. Math. Soc. 131

(2002), 441–448.

[2] J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrodinger equations II. J. Differential

Equations 187, (2003), 473–493.

[3] X. Liu; J. Liu and Z. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc.,

141 (2013), 253–263.


THE IVP FOR THE EVOLUTION EQUATION OF WAVE FRONTS IN CHEMICAL REACTIONS

IN LOW-REGULARITY SOBOLEV SPACES

ALYSSON CUNHA1

1Instituto de Matematica e Estatıstica, IME-UFG, GO, Brasil, [email protected]

Abstract

In this work, we study the initial-value problem for an equation of evolution of wave fronts in chemical

reactions. We show that the associated initial value problem is locally and globally well-posed in Sobolev spaces

Hs(R), where s > 1/2. The well-posedness in critical space H1/2(R), for small initial data is obtained. We also

show that our result is sharp, in the sense that the flow-map data-solution is not C2 at origin, for s < 1/2.

Furthermore, we study the behavior of the solutions when µ ↓ 0.

1 Introduction

This work is concerned with the initial-value problem (IVP), for the evolution equation of wave fronts in chemical

reactions (WFCR) ut − ∂2xu− µ(1 − ∂2x)−1/2u− 12 (∂xu)2 = 0, x ∈ R, t ≥ 0,

u(x, 0) = ϕ(x),(1)

where above µ > 0 is a constant, u is a real-valued function and the operator (1 − ∂2x)−1/2 is defined via your

Fourier transform by

((1 − ∂2x)−1/2f)∨ = (1 + ξ2)−1/2f(ξ).

The IVP (1) describe vertical propagation of chemical waves fronts in the presence instability due to density

gradients.

2 Main Results

In the following, we show our main results, see [3].

Theorem 2.1. (Local well-posedness). Let µ > 0 and s > 1/2, then for all ϕ ∈ Hs(R), there exists T = T (∥ϕ∥Hs),

a space

X sT → C([0, T ];Hs(R))

and a unique solution u of (1) in X sT . In addition, the flow map data-solution

S : Hs(R) → X sT ∩ C([0, T ];Hs), ϕ 7→ u

is smooth and

u ∈ C((0, T ];H∞(R)).

Theorem 2.2. Let µ > 0 and 0 < T ≤ 1. If ϕ ∈ H1/2(R) is such that ∥ϕ∥H1/2 < (4kCµ)−1, then there exists a

space

X 1/2T → C([0, T ];H1/2(R)),

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and a unique solution u of (1) in X 1/2T . In addition, the flow map data-solution

S : H1/2(R) → X 1/2T ∩ C([0, T ];H1/2), ϕ 7→ u

is smooth and

u ∈ C((0, T ];H∞(R)).

For the next result, Hs denotes the homogeneous Sobolev space. The constants k and Cµ, in the next results,

depend on s, T and µ.

Theorem 2.3. If the initial data is such that ∥ϕ∥H1/2 < (4kCµ)−1, then the IVP (1) is locally well-posed in H1/2.

Theorem 2.4. (Global well-posedness). Let µ > 0 and s > 1/2, then the initial value problem (1) is globally

well-posed in Hs(R).

Theorem 2.5. (Ill-posedness). Let s < 1/2, if there exists some T > 0, such that the problem (1) is locally

well-posed in Hs(R), then the flow-map data solution

S : Hs(R) → C([0, T ];Hs(R)), ϕ 7→ u,

is not C2 at zero.

To obtain the above results, we use techniques present in [2].

References

[1] carvajal, x. and panthe, m. - Sharp local well-posedness of Kdv type equations with Dissipative

perturbations. Quarterly of Applied Mathematics., 74, 571–594, 2016.

[2] cunha, a. and alarcon, e. - The IVP for the evolution equation of wave fronts in chemical reactions in

low-regularity Sobolev spaces. J. Evol. Equ., 21, 921–940, 2021.


EXISTENCIA DE SOLUCOES PERIODICAS EM ESCOAMENTOS DE FERROFLUIDOS

JAUBER C. OLIVEIRA1

1Departamento de Matematica, UFSC, SC, Brasil, [email protected]

Abstract

Neste trabalho apresentamos resultados sobre a existencia de solucoes fortes periodicas no tempo para um

sistema de equacoes diferenciais parciais associadas a um modelo (de Shliomis) bidimensional e tridimensional

para o escoamento de fluidos ferro-magneticos. O regime periodico no tempo e induzido por um campo magnetico

externo. No caso tridimensional, supomos que o campo magnetico externo e suficientemente pequeno em

determinada norma.

1 Introducao

Escoamentos de fluidos magneticos ([2]) aparecem em varias aplicacoes industriais ([1]). Este estudo foi motivado

pelo interesse em aplicacoes em que deseja-se induzir escoamentos de fluıdos magneticos em regime periodico no

tempo por meio de um campo magnetico externo.

O modelo considerado e o modelo de Shliomis, representado pelo seguinte sistema de equacoes diferenciais

parciais.

∇ · u = 0 em Ω × (0,∞), (1)

ρ (ut + (u · ∇)u) − η u + ∇p = µ0(m · ∇)h +µ0

2∇× (m ∧ h) em Ω × (0,+∞), (2)

ρ (ut + (u · ∇)u) − η u + ∇p = µ0(m · ∇)h +µ0

2∇× (m ∧ h) em Ω × (0,+∞), (3)

mt + (u · ∇)m− σ∆m =1

2rotu ∧m− 1

τ(m− χ0h) − βm ∧ (m ∧ h) em Ω × (0,+∞), (4)

∇ · h = −∇ ·m + F, ∇× h = 0 em Ω × (0,+∞). (5)

Estas equacoes descrevem o balanco de massa, de momento linear e de magnetizacao, respectivamente. u denota

a velocidade do ferro-fluıdo, p denota a pressao dinamica, h representa o campo magnetico e m e a magnetizacao.

ρ, η, µ0, τ, χ0, β sao constantes positivas. O modelo descreve o escoamento de um fluıdo magnetico sujeito a um

campo magnetico externoHext, com F = −∇·Hext. Se o termo regularizante −σ∆m e desconsiderado (desprezando-

se o momento magnetico de rotacao [3]), ate mesmo a existencia de solucoes fracas nao e conhecida. As condicoes

de contorno e de periodicidade sao as seguintes:

u = 0, m · ν = 0, ∇×m ∧ ν = 0 on ∂ΩT , (6)

u(0) = u(T ), m(0) = m(T ), h(0) = h(T ) em Ω. (7)

Os seguintes trabalhos anteriores abordam questoes de existencia de solucoes para este modelo: [3],[5],[1],[7].

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Sejam

V(Ω) := φ ∈ D(Ω) : divφ = 0 , Hdiv(Ω) :=u ∈ L2(Ω) : divu ∈ L2(Ω)

com norma

∥u∥Hdiv:=(∥u∥2 + ∥div u∥2

)1/2.

V (Ω) e o fecho de V(Ω) em H10 (Ω). Hdiv,0(Ω) e o fecho de D(Ω) em Hdiv(Ω).

Seja A o operador de Stokes, A : V (Ω) → V (Ω)∗ definido por ⟨Av,u⟩V ∗,V =∫Ω∇u : ∇v dx, ∀u,v ∈ V (Ω) com

domınio D(A) := u ∈ V (Ω) : Au ∈ H(Ω) e (u|v)D(A) := (u|v) + (Au|Av), ∀u,v ∈ D(A).

Temos a seguinte caracterizacao para o espaco Hν,0(Ω) = H1(Ω) ∩ N (γν), onde γν(u) = u · ν e o operador traco,

contınuo de Hdiv(Ω) em H−1/2(∂Ω). Seja L( ) = −∆( ) com domınio D(L) = H2(Ω) ∩Hdiv,0(Ω).

Apresentamos neste trabalho os seguintes resultados de regularidade das solucoes fracas. O existencia de solucoes

fracas esta entre os resultados desta investigacao. Apresentamos somente resultados de regularidade destas solucoes.

Teorema 2.1. (Regularidade de solucoes fracas - caso 2d)

Seja T > 0 o perıodo da funcao F ∈ H1(0, T ;L2(Ω)) tal que (F |1) = 0 em [0, T ]. Seja Ω ⊂ R2 um conjunto aberto

limitado com fronteira regular (pelo menos de classe C3). Entao, as solucoes fracas T -periodicas (u,m,h) tem a

seguinte regularidade adicional:

u ∈ L∞(0, T ;V (Ω)) ∩ L2(0, T ;D(A)), m ∈ L∞(0, T ;Hν,0(Ω)) ∩ L2(0, T ;D(L)), e

h ∈ L∞(0, T ;Hν,0(Ω)).

Alem disso, se F ∈ L2(0, T ;H1(Ω)), entao h ∈ L2(0, T ;H2(Ω)).

Teorema 2.2. (Regularidade de solucoes fracas - caso 3d)

Seja F ∈ H1(0, T ;H1(Ω)) tal que (F |1) = 0 em [0, T ]. Ω ⊂ R3 e um subconjunto aberto, limitado, simplesmente

conexo com fronteira suave (pelo menos de classe C3). Existe uma constante c0 tal que se ∥F∥H1(0,T ;H1(Ω)) ≤ c,

entao as solucoes fracas T -periodicas (u,m,h) tem a seguinte regularidade adicional:

u ∈ L∞(0, T ;V (Ω)) ∩ L2(0, T ;D(A)), m ∈ L∞(0, T ;Hν,0(Ω)) ∩ L2(0, T ;D(L))

h ∈ L∞(0, T ;Hν,0(Ω)) ∩ L2(0, T ;H2(Ω)).

References

[1] odenbach, s. (Ed.) - Colloidal Magnetic Fluids: Basics, Development and Application of Ferrofluids, Lect.

Notes Phys. 763, Springer, Berlin Heidelberg, 2009.

[2] rosensweig, r.e. - Ferrohydrodynamics, Dover Publications, 2014.

[3] torrey, h.c. - Bloch Equations with Diffusion Terms, Phys. Rev., 104(3) (1956): 563-265.

[4] amirat, y., hamdache, k. - Global Weak Solutions to a Ferrofluid Flow Model, Math. Meth. Appl. Sci., 31

(2008): 123-151.

[5] amirat, y., hamdache, k. - Strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl.

353 (2009): 271-294.

[6] oliveira, j.c. - Strong solutions for ferrofluid equations in exterior domains, Acta Appl. Math., 156 (2018):

1-14.

[7] xie, c. - Global strong solutions to the Shliomis system for ferrofluids in a bounded domain, Math Meth. Appl.

Sci. (2019): 1-8.


EXACT BOUNDARY CONTROLLABILITY FOR THE WAVE EQUATION IN MOVING

BOUNDARY DOMAINS WITHA STAR-SHAPED HOLE

RUIKSON S. O. NUNES1

1Departamento de Matematica, UFMT, MT, Brasil, [email protected]

Abstract

We consider an exact boundary control problem for the wave equation in a moving bounded domain which

has a star-shaped hole. The boundary domain is composed by two disjoint parts, one is the boundary of the

hole, which is fixed, and the other one is the external boundary which is moving. The initial data has finite

energy and the control obtained is square integrable and is obtained by means of the conormal derivative. We

use the method of controllability presented by Russell in [2], and assume that the control acts only in the moving

part of the boundary.

1 Introduction

In this work we study an exact boundary control problem for the standard wave equation on a domain with moving

boundary which has a single fixed hole. The boundary of such domains is composed by two disjoint parts: one

it is the boundary on hole which is fixed, and the other one is the external boundary which is moving. We shall

consider the control acting only on the moving boundary part. A illustrative example would be a flexible body that

is crossed by a cylindrical pillar and is fixed to it. Without any variation in the temperature of the environment

the body has no dilation and thus its external boundary remains static. However, if there is a variation in the

temperature, the body would have a dilation or a contraction, causing the mobility of the its external boundary.

In this work when we deal with a domain with a hole, and we refer by external boundary as being the part of the

boundary of the domain that does not coincide with that one of the hole.

To establish these concepts in more detail, we consider B ⊂ Rn, n ≥ 2, a convex compact set with the origin in

its interior with smooth boundary Γ0. We set Ω∞ = Rn −B. Let Ξ ⊂ Rn be a simply connected bounded domain

with piecewise smooth boundary Γ1, with no cusps, such that B ⊂ Ξ. We assume that dist(Γ0,Γ1) ≥ ϵ > 0 and set

Ω = Ξ−B. Hence, the boundary of Ω is ∂Ω = Γ0 ∪Γ1. Note that Ω is a holed domain whose hole has the shape of

B. We also consider the moving boundary domain Ξt ⊂ Rn where Ξt = x ∈ Rn : x = α(t)y, y ∈ Ξ, t ∈ [0,+∞)

whose boundary is denoted by Γt and Ξ0 = Ξ. Here α : R+ ∪ 0 → R is a piecewise bounded smooth function,

where

Ξt × R ⊂ ∪x∈Ξ(x, t) ∈ RN × R : |x− x|2 ≤ t2, (1)

Ξt ⊂ B(0, r) for all t ∈ [0,+∞). Defining Ω = B(0, r) − B and Ωt = Ξt − B we can see that Ωt ⊂ Ω for all

t ∈ [0,+∞). The boundary of Ωt is ∂Ωt = Γ0 ∪ Γt. Now, for T > 0, let us consider the non-cylindrical domain of

Rn+1, QT = ∪0<t<T Ωt×t whose the lateral boundary is ΣT ∪Σ0, where ΣT = ∪0<t<T Γt×t and Σ0 = Γ0×[0, T ].

We denote by (νx, νt) the outward unit normal vector defined almost all on ΣT ∪ Σ0. Note that QT is a holed

non-cylindrical domain in Rn+1 whose the lateral boundary is composed by two disjoint parts ΣT and Σ0. Here,

we requires that B be star-shaped with respect the origin, that is, νx · x ≤ 0 for x ∈ ∂B. The assumption (1)

assures that the surface ΣT is time-like. This is known to be sufficient to guarantee the well-posedness of the initial

and boundary value problem studied here. The purpose of this work is to study the exact boundary controllability

problem

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Theorem 1.1. Let Ω be as defined above. Given (f, g) ∈ H1(Ω) × L2(Ω), there exist T > 0 sufficiently large and

a control function h(·, t) ∈ L2(ΣT ) such that the solution u ∈ H1(QT ) of the problem

utt − ∆u = 0 in QT

u(·, 0) = f, ut(·, 0) = g, in Ω

u(·, t) = 0, on Γ0 × [0, T ]

νtut −∇u · νx = h(·, t), on ΣT .

(2)

satisfy the final condition

u(·, T ) = 0 = ut(·, T ) in ΩT . (3)

2 Idea of the proof of the Theorem 1

The Theorem 1 is proved completely in the paper [1]. Here we show the sketch of the its proof. Firstly we the

solution u of the initial-boundary value problem

utt − ∆u = 0 in Ω∞ × (0,+∞)

u(·, 0) = f , ut(·, 0) = g, in Ω∞

u(·, t) = 0, in Γ0 × (0,+∞).

(4)

Being f and g extension of f and g respectively to Ω∞. After, for a T > 0 great sufficiently we take the state

(u(·, T ), ut(·, T )) ∈ H1(Ω) × L2(Ω) and we solve the exact boundary control problem

vtt − ∆v = 0 in Ω × [0, T ]

v(·, T ) = u(·, T ), vt(·, T ) = ut(·, T ), in Ω

v(·, t) = 0, on Γ0 × [0, T ]

νtvt −∇v · νx = h(·, t), on Γ × [0, T ],

(5)

which satisfies, at the instant t = 0, the condition v(·, 0) = 0 = vt(·, 0) in Ω.

Considering Ωt as defined in the Section 1, note that Ωt ⊂ Ω for all t > 0, so follows that for each T > 0 we

have QT = ∪0<t<T Ωt × t ⊂ Ω × [0, T ]. Defining u = u− v we can see that the restriction of u to QT satisfies (2)

and the condition (3). The control function h is obtained by means of a trace theorem established in [3].

References

[1] R. S. O. Nunes;- Exact boundary controllability for the wave equation with moving boundary domains in a

star-shaped hole, Electron. J. Differential Equations, 2021 No. 49, (2021) 1–12.

[2] D. L. Russell;-Exact boundary value controllability theorems of wave and heat processes in star-complemented

regions, Differential Games and Control Theory, Roxin, Liu and Sternberg, ed., Marcel Dekker, New York,

1974.

[3] D. Tataru, On regularity of the boundary traces for the wave. Ann. Scuola Norm. Pisa, C. L. Sci.(4) 26 (1),

(1998) 185 - 206.


MAXIMAL ATTRACTORS FOR SEMIGROUPS

MATHEUS C. BORTOLAN1

1Departamento de de Matematica, UFSC, SC, Brasil, [email protected]

Abstract

The theory of compact global attractors for semigroups relies on the existence of a bounded absorbing

set. Here, suppressing this condition, we present a general theory of maximal attractors for semigroups. Such

attractors are, in general, unbounded. We present sufficient conditions to ensure their existence. As an example

we present a semilinear parabolic equation.

This is a joint work with Juliana Fernandes (UFRJ), and the results presented are extracted from our

submitted paper [1].

1 Introduction and Main Results

For the last several decades, semigroups with bounded absorbing sets have been extensively studied by many

authors. However, if unbounded solutions exist, the system has no such bounded absorbing set. A simple example

is the semigroup generated by a linear map in R2 with one eigenvalue outside and the other inside the unitary circle,

in which case the solutions converge to the line defined by the eigenvector associated with the eigenvalue outside

the unitary circle. Despite the fact that unbounded attractors are quite common objects in evolution equations,

they are harder to study, due to the lack of compactness, and much less is known in their regard. In this work, we

explore the asymptotic behavior of semigroups without assuming the existence of a bounded absorbing set.

We explain now our main results. In a metric space X, we consider a semigroup T = T (t) : t ⩾ 0. A maximal

attractor for T is a closed subset U of X that satisfies:

(i) U is invariant, that is, T (t)U = U for all t ⩾ 0;

(ii) U attracts bounded subsets of X, that is, for each bounded subset B of X we have distH(T (t)B,U)t→∞−−−→ 0,

where distH(C,D) = supc∈C

infd∈D

d(c, d) denotes the Hausdorff semidistance between two nonempty subsets C,D

of X;

(iii) there is no proper closed subset V of U that satisfies both (i) and (ii).

The task to find a maximal attractor for a semigroup is arduous, and not possible in general. However, following

the ideas of Chepyzhov and Goritskiı [2], we find conditions under which the set of bounded in the past global

solutions of a semigroup T , given by J = ξ(0) : ξ is a bounded in the past global solution of T, is a maximal

attractor for T . The main result of our work is as follows:

Theorem 1.1. Let (X, d) be a complete metric space and T be a semigroup in X with the set of bounded in the

past global solutions J nonempty. Assume that:

(a) T is asymptotically compact, that is, given a bounded subset B of X there exists t0 = t0(B) ⩾ 0 such that

for each t ⩾ t0 there exist a compact subset K = K(B, t) of X and ϵ = ϵ(B, t) ⩾ 0 with distH(T (t)B,K) < ϵ,

with ϵt→∞−−−→ 0;

(b) T has a strongly absorbing set G, which means that:

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82

(H1) G is positively invariant, that is, T (t)G ⊂ G for all t ⩾ 0;

(H2) for each bounded subset B of X, there exists t0 = t0(B) ⩾ 0 such that T (t)B ⊂ G for all t ⩾ t0;

(H3) there exists a sequence of bounded subsets (Hn)n∈N of G with the following properties:

⋆ Hn ⊂ Hn+1 for all n ∈ N;⋆ G \Hn is positively invariant for T for each n ∈ N;⋆ if B ⊂ G is bounded, then B ⊂ Hn for some n ∈ N.

(H4) distH(T (t)G,J )t→∞−−−→ 0.

Then J is the unique maximal attractor for T , and J is bounded-compact, that is, the intersection of J with

each closed bounded subset of X is compact.

To illustrate the theory we use Theorem 1.1 to obtain that J is the bounded-compact maximal attractor for

the semigroup generated by the following n-dimensional semilinear parabolic equation:ut = ∆u+ bu+ f(u), in (0,∞) × Ω

u = 0, in (0,∞) × ∂Ω,

u(0) = u0, in Ω,

(1)

where Ω ⊂ Rn is a bounded domain (open and connected) with sufficiently smooth boundary, b > 0, u0 ∈ L2(Ω)

and f : R → R is a C2 function satisfying

|f(r)| ⩽ c and∣∣∣f ′(r)∣∣∣ ⩽ c for all r ∈ R, (2)

for some given constant c ⩾ 0.

References

[1] bortolan, m.c. and fernandes, j. - Maximal attractors for autonomous and nonautonomous dynamical

systems, Submitted for publication, 2021.

[2] chepyzhov, v.v. and goritskiı, a.y. - Unbounded attractors of evolution equations, Properties of global

attractorsof partial differential equations, Adv. Soviet Math., 10, 85-128, 1992.


CONTROLLABILITY UNDER POSITIVE CONSTRAINTS FOR QUASILINEAR PARABOLIC

PDES

MIGUEL R. NUNEZ CHAVEZ1

1Instituto de Matematica e Estatıstica, UFF, Niteroi-RJ, Brasil, [email protected]

Abstract

In this work, we deals with the analysis of the internal control with constraint of positive kind of a parabolic

PDE with nonlinear diffusion when the time horizon is large enough. The minimal controllability time will be

strictly positive.

We prove a global steady state constrained controllability result for a quasilinear parabolic with nonlinearity

in the diffusion term. Then, under suitable dissipative assumption in the system and local controllability results,

we conclude the result to any initial datum and any target trajectory.

1 Introduction

Let Ω ⊂ RN (N ≥ 1 is an integer) be a non-empty bounded connected open set, with regular boundary ∂Ω. We fix

T > 0 and set Q := Ω × (0, T ) and Σ := ∂Ω × (0, T ).

Let ω, ω1 ⊂ Ω be non-empty open sets, such that ω1 ⊂⊂ ω. We deal with the exact controllability to trajectories

for the quasilinear system yt −∇ · (a(y)∇y) = vϱω in Q,

y(x, t) = 0 on Σ,

y(x, 0) = y0(x) in Ω,

(1)

where y is the associated state, v is the control and ϱω ∈ C∞0 (Ω), such that ϱω = 0 in Ω\ω and ϱω = 1 in ω1.

Here, it will be assumed that the real-valued function a = a(r) satisfies

a ∈ C2(R), 0 < a0 ≤ a(r) and |a′(r)| + |a′′(r)| ≤M, ∀r ∈ R. (2)

Definition 1.1. Let v ∈ C1/2(Ω), a function y ∈ C2+1/2(Ω) is said to be a steady state for (1) if it is a solution

to

−∇ · (a(y)∇y) = vϱω in Ω, y = 0 in ∂Ω. (3)

The function v ∈ C1/2(Ω) is called the steady control.

Remark 1.1. The application Λ : v 7→ y shown in (3) is continuous, since a(·) satisfies (2).

We will denote by S := Λ(C1/2(Ω)) the set of all the steady-states with steady controls in C1/2(Ω).

Definition 1.2. Fixed y0, y1 ∈ S and fixed v0, v1 such that Λ(v0) = y0 and Λ(v1) = y1, we define a path-

connected steady states that drive y0 to y1 as a continuous path

γ : [0, 1]λ−→ C1/2(Ω)

Λ−→ S

s 7−→ λ(s) 7−→ γ(s) = Λ(λ(s)),

where λ(s) is a continuous path of steady controls that drive v0 to v1 (λ(0) = v0 and λ(1) = v1).

For each s ∈ [0, 1], we denote ys := γ(s) the steady state and vs := λ(s) the steady control of continuous path γ .

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84

Definition 1.3. Let us define a target trajectory y = y(x, t) for (1) as solution toyt −∇ · (a(y)∇y) = vϱω in Q,

y(x, t) = 0 on Σ,

y(x, 0) = y0(x) in Ω,

(4)

with y0 ∈ C2+1/2(Ω) and v ∈ C1/2,1/4(Q) such that

Ma∥∇y∥L∞(Ω×(0,T )) ≤a0

2 C(Ω), (5)

where C(Ω) is the Poincare inequality constant, so ∥u∥L2(Ω) ≤ C(Ω)∥∇u∥L2(Ω) and the constant Ma is defined by

Ma := supr∈R

|a′(r)|.

2 Main Results

Now, let us state the main result in this section is the following

Theorem 2.1. Let y0, y1 ∈ S fixed and let γ(s) := ys be path-connected steady states that drive y0 to y1 with steady

control vs. Let us assume there exists a constant η > 0 such that

vs ≥ η, ∀s ∈ [0, 1]. (1)

Then there exists T0 > 0 such that, for every T ≥ T0 there exists a control v ∈ L∞(Ω× (0, T )) such that, the system

(1) admits a unique solution y satisfying y(·, T ) = y1(·) in Ω and v ≥ 0 in Ω × (0, T ).

Now, we will extend Theorem 2.1 in the following Theorem:

Theorem 2.2. Suppose there exists a target trajectory y satisfying the condition (5) with initial datum y0 and

control v. Let us assume there exist a constant η > 0 such that

v ≥ η in Ω × R+. (2)

For any y0 ∈ C2+1/2(Ω) initial datum, there exists T0 > 0 such that for every T ≥ T0, we can find a control

v ∈ L∞(Ω × (0, T )) such that the unique solution y to (1) satisfies y(T ) = y(T ) in Ω and v ≥ 0 in Ω × (0, T ).

Furthermore, if y0 = y0 then the minimal controllability time Tmin is strictly positive, where

Tmin := inf T > 0; ∃ v ∈ L∞(Ω × (0, T ))+, such that y(T ) = y(T ) in Ω. (3)

References

[1] Ladyzhenskaya O. A.; Solonnikov V. A.; Ural’ceva N. N. Linear and Quasilinear Elliptic Equations,

Translations by Scripta Technica, Inc, Academy Press, New York and London (1968).

[2] Ladyzhenskaya O. A.; Solonnikov V. A.; Ural’ceva N. N. Linear and Quasilinear Equations of Parabolic Type,

Translations of Mathematical Monographs, vol. 23, AMS, Providence, RI, (1968).

[3] Liu X.; Zhang X. Local Controllability of Multidimensional Quasi-Linear Parabolic Equations, SIAM Journal

on Control and Optimization, 50(4), (2012), 2046-2064.

[4] Loheac J.; Trelat E.; Zuazua E. Minimal controllability time for the heat equation under unilateral state or

control constraints, Mathematical Models and Methods in Applied Sciences, 27(9), (2017), 1587-1644.

[5] Pighin D.; Zuazua E. Controllability under positive constraints of Semilinear Heat Equations, Mathematical

Control and Related Fields, 8(34), (2018), 935-964.


ON A VARIATIONAL INEQUALITY FOR A BEAM EQUATION WITH INTERNAL DAMPING

AND SOURCE TERMS

GERALDO M. DE ARAUJO1 & DUCIVAL C. PEREIRA2

1Instituto de Ciencias Exatas e Naturais, Faculdade de Matematica, UFPA, PA, Brasil, [email protected],2Departamento de Matematica, UEPA, PA, Brasil, [email protected]

Abstract

In this paper we investigate the unilateral problem for a extensible beam equation with internal damping

and source terms

utt +∆2u+M(|∇u|2)(−∆u) + ut = |u|r−1u

where r > 1 is a constant, M(s) is a continuous function on [0,+∞). The global solutions are constructed

by using the Faedo-Galerkin approximations, taking into account that the initial data is in appropriate set of

stability created from the Nehari manifold.

1 Introduction

In [7] the authors establish existence of global solution to the problem

utt + ∆2u+M(|∇u|2)(−∆u) + ut = |u|r−1u (1)

u(., 0) = u0, ut(., 0) = u1 in Ω, (2)

u(., t) =∂u

∂η(., t) in ∂Ω, t ≥ 0, (3)

where Ω is a bounded domain of Rn with smooth boundary ∂Ω , r > 1 is a constant and M(s) is a continuous

function on [0,+∞), u = 0 is the homogeneous Dirichlet boundary condition and the normal derivative∂u

∂η= 0 is

the homogeneous Neumann boundary condition, where η unit outward normal on ∂Ω.

A nonlinear perturbation of problem (1) is given by

utt + ∆2u+M(|∇u|2)(−∆u) + ut − |u|r−1u ≥ 0. (4)

In the present work we investigated the unilateral problem associated with this perturbation, that is, a variational

inequality given for (4) (see [5]). Making use of the penalty method, the potential well theory and Galerkin’s

approximations, we establish existence and uniqueness of global solutions.

Unilateral problem is very interesting too, because in general, dynamic contact problems are characterized

by nonlinear hyperbolic variational inequalities. For contact problem on elasticity and finite element method see

Kikuchi-Oden [4] and reference there in. For contact problems on viscoelastic materials see [6]. For contact problems

on Klein-Gordon operator see [8]. For contact problems on Oldroyd Model of Viscoelastic fluids see [3]. For contact

problems on Navier-stokes Operator with variable viscosity see [1]. For contact problems on viscoelastic plate

equation see [1].

85

86

2 Main Results

Theorem 2.1. Consider the spaces

H4Γ(Ω) = u ∈ H4(Ω)|u = ∆u = 0 on Γ and H3

Γ(Ω) = u ∈ H3(Ω)|u = ∆u = 0 on Γ.If u0 ∈W1 ∩H4

Γ(Ω), J(u0) < d, u1 ∈ H10 (Ω)∩H2(Ω), 1 < r ≤ 5 and the hypothesis (H1) and (H2) holds, then there

exists a function u : [0, T ] → L2(Ω) in the class

u ∈ L∞(0, T ; (H10 (Ω) ∩H2(Ω)) ∩H3

Γ(Ω)) ∩ L∞(0, T ;Lr+1(Ω)) (1)

ut ∈ L∞(0, T ;L2(Ω)) ∩ L2(0, T ;H10 (Ω) ∩H2(Ω)) (2)

utt ∈ L∞(0, T ;L2(Ω)), ut(t) ∈ K a.e. in [0, T ], (3)

satisfying∫Q

(utt + ∆2u+M(|∇u|2)(−∆u) + ut − |u|r−1u)(v − ut) ≥ 0,∀v ∈ L2(0, T ;H10 (Ω)), v(t) ∈ K a.e. in t

u(0) = u0, ut(0) = u1

(4)

where J : H10 (Ω) ∩H2(Ω) → R defined by J(u) =

1

2|∆u|2 +

m0

2|∇u|2 − 1

r + 1|u|r+1

r+1, and

(H1) M ∈ C1([0,∞)) with M(λ) ≥ m0 > 0, ∀λ ≥ 0, (H2) (r − 1)n ≤ rn ≤ q =2n

n− 2, n > 2.

Proof The proof of Theorem 2.1 is made by the penalization method. It consists in considering a perturbation of

the problem (1) adding a singular term called penalty, depending on a parameter ϵ > 0. We solve the mixed problem

in Q for the penalization operator and the estimates obtained for the local solution of the penalized equation, allow

to pass to limits, when ϵ goes to zero, in order to obtain a function u which is the solution of our problem.

References

[1] G. M. Araujo, M. A. F. Araujo and D. C. Pereira (2020), On a variational inequality for a plate equation with

p-Laplacian and memory terms, Applicable Analysis, DOI: 10.1080/00036811.2020.1766028.

[2] G. M. De Araujo and S. B. De Menezes On a Variational Inequality for the Navier-stokes Operator with

Variable Viscosity, Communications on Pure and Applied Analysis. Vol. 1, N.3, 2006, pp.583-596.

[3] G. M. De Araujo, S. B. De Menezes and A. O. Marinho On a Variational Inequality for the Equation of Motion

of Oldroyd Fluid, Electronic Journal of Differential Equations, Vol. 2009(2009), No. 69, pp. 1-16.

[4] Kikuchi N., Oden J. T., Contacts Problems in Elasticity: A Study of Variational inequalities and Finite

Element Methods. SIAM Studies in Applied and Numerical Mathematics: Philadelphia, (1988).

[5] J.L. Lions, Quelques Methodes de Resolution Des Problemes Aux Limites Non Lineaires, Dunod, Paris, 1969.

[6] Munoz Rivera J.A., Fatori L. H. Smoothing efect and propagations of singularities for viscoelastic plates.

Journal of Mathematical Analysis and Applications 1977; 206: 397-497.

[7] D. C. Pereira, H. Nguyen, C.A. Raposo and C. H. M. Maranhao,On the solutions for an extensible beam

equation with internal damping and source term, Differential Equations and Applications, V. 11, N. 3 (2019),

367-377.

[8] Raposo C. A., Pereira D. C., Araujo G., Baena A., Unilateral Problems for the Klein-Gordon Operator with

nonlinearity of Kirchhoff-Carrier Type, Electronic Journal of Differential Equations, Vol. 2015(2015), No. 137,

pp. 1-14.


THE NONLINEAR QUADRATIC INTERACTIONS OF THE SCHRODINGER TYPE ON THE

HALF-LINE

ISNALDO ISAAC BARBOSA1 & MARCIO CAVALCANTE2

1IM, UFAL, Maceio (AL), Brazil, [email protected],2IM, UFAL, Maceio (AL), Brazil, [email protected]

Abstract

In this work we study the initial boundary value problem associated with the coupled Schrodinger equations

with quadratic nonlinearities, that appears in nonlinear optics, on the half-line. We obtain local well-posedness

for data in Sobolev spaces with low regularity, by using a forcing problem on the full line with a presence of a

forcing term in order to apply the Fourier restriction method of Bourgain. The crucial point in this work is the

new bilinear estimates on the classical Bourgain spacesXs,b with b < 12, jointly with bilinear estimates in adapted

Bourgain spaces that will used to treat the traces of nonlinear part of the solution. Here the understanding of

the dispersion relation is the key point in these estimates, where the set of regularity depends strongly of the

constant a measures the scaling-diffraction magnitude indices.

This work was submitted for publication and can be accessed in https://arxiv.org/abs/2104.05137 .

1 Introduction

This work is dedicated to the study the initial boundary value problem associated to system nonlinear quadratic

of the Schrodinger on the half-line, more preciselyi∂tu(x, t) + ∂2xu(x, t) + u(x, t)v(x, t) = 0, x ∈ (0,+∞), t ∈ (0, T ),

i∂tv(x, t) + a∂2xv(x, t) + u2(x, t) = 0, x ∈ (0,+∞), t ∈ (0, T ),

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ (0,+∞),

u(0, t) = f(t), v(0, t) = g(t), t ∈ (0, T ),

(1)

where u and v are complex valued functions, where a > 0. The model (1) is given by the nonlinear coupling of two

dispersive equations of Schrodinger type through the quadratic terms N1(u, v) = u · v and N2(u, v) = u2.

An important point in this model is the fact that the functional mass is not conserved, since some bad terms of

boundary appear in the mass functional. More precisely, define the functional of mass for the system (1) by

M(t) = ∥u(t)∥2L2x(R+) + ∥v(t)∥2L2

x(R+).

Formally, by multiplying the first equation of the system (1) by u and the second equation by v, integrating by

parts, taking the imaginary part and using Im (u2v) = −Im(u2v), we get

M(t) = M(0) + Im

∫ t

0

u(0, s)∂xu(0, s)ds+ aIm

∫ t

0

v(0, s)∂xv(0, s)ds. (2)

This identity suggesters on the case of homogeneous boundary conditions a global result on the space

L2(R+) × L2(R+).

Physically, according to the article [4], the complex functions u and v represent amplitude packets of the first

and second harmonic of an optical wave. In the mathematical context on the paper [1] the first author obtained

local well posedness for the model posed on real line by assuming low regularity assumptions.

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88

2 Main Results

Our main local well-posedness result is the following statement.

Theorem 2.1. Let the Sobolev index pair (κ, s) verifying s = 12 and κ = 1

2 and

(i) |κ| − 1/2 ≤ s < minκ+ 1/2, 2κ+ 1/2, 1 and κ < 1 for a > 12 (first non resonant case);

(ii) 0 ≤ κ = s < 1 for a = 12 (resonant case);

(iii) max− 12 , |κ| − 1 ≤ s < minκ + 1, 2κ + 1, 1 and κ < 1 for 0 < a < 1

2 (second non resonant case). For

any a > 0 and (u0, v0) ∈ Hκ(R+) ×Hs(R+) and (f, g) ∈ H2κ+1

4 (R+) ×H2s+1

4 (R+), verifying the additional

compatibility conditions

u(0) = f(0), for κ > 12 ;

v(0) = g(0), for s > 12 .

(3)

Then there exist a positive time T = T

(∥u0∥Hκ(R+), ∥v0∥Hs(R+), ∥f∥

H2κ+1

4 (R+), ∥g∥

H2s+1

4 (R+), a

)and a

distributional solution (u(t), v(t)) for the initial boundary value problem (1) on the classes

u ∈ C([0, T ];Hκ(R+)

)and v ∈ C

([0, T ];Hs(R+)

). (4)

Moreover, the map (u0, v0) 7−→ (u(t), v(t)) is locally Lipschitz from Hκ(R+) ×Hs(R+) into

C ([0, T ];Hκ(R+) ×Hs(R+)).

The approach used to prove this result is based on the arguments introduced in [3] and [2]. The main idea to

solve the IBVP (1) is the construction of an auxiliary forced IVP in the line R, analogous to (1); more precisely:i∂tu(x, t) + ∂2xu(x, t) + u(x, t)v(x, t) = T1(x)h1(t), (x, t) ∈ R× (0, T )

i∂tv(x, t) + a∂2xv(x, t) + u2(x, t) = T2(x)h2(t), (x, t) ∈ R× (0, T )

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ R(5)

where T1, T2 are appropriate distributions supported in R−, u0, v0 are nice extensions of u0 and v0 in R and the

boundary forcing functions h1, h2 are selected to ensure that

u(0, t) = f(t) and v(0, t) = g(t)

for all t ∈ (0, T ).

References

[1] I. Barbosa. The Cauchy problem for nonlinear quadratic interactions of the Schrodinger type in one

dimensional’space. Journal of Mathematical Physics 59(7): 2018

[2] M. Cavalcante, The initial-boundary value problem for some quadratic nonlinear Schrodinger equations on the

half-line, Differential Integral Equations 30 (7-8) (2017) 521-554.

[3] J. E. Colliander, C. E. Kenig,: The generalized Korteweg-de Vries equation on the half line, Comm. Partial

Differential Equations, 27 (2002), no. 11/12, 2187–2266.

[4] C. Menyuk, R. Schiek, and L. Torner. Solitary waves due to χ (2): χ (2) cascading. JOSA B, 11(12):2434–2443,

1994.


VIBRATIONS OF A BAR SUBMITTED TO AN IMPACT

M. MILLA MIRANDA1, L. A. MEDEIROS2 & A. T. LOUREDO3

1Departamento de Matematica, Campina Grande, UEPB, PB, Brasil, [email protected],2Instituto de Matematica, UFRJ, RJ, Brasil, [email protected],

3Departamento de Matematica, Campina Grande, UEPB, PB, Brasil, [email protected]

Abstract

In this paper is investigated the existence of solutions of a mathematical model that describes the vibrations

of a bar by an impact in one of its ends.

1 Introduction

Consider an elastic homogeneous cylindrical bar of lenght L where the cross sections of the bar are small when

comparing with its lenght. In the rest position the bar coincides whith the interval [0.L] of the axis Ox. At the

end x = 0, the bar is clamped and the end x = L is free. At the initial time t = 0, the free end is hit by a mass

M , which is moving with velocity α0 in the direction of the axis of the bar. Then the mass remains glued at the

end x = L. Under the impact, the cross sections of the bar begin to vibrate longitudinally. Assume that these

vibrations are small.

The above physical problem is modeled by the following mathematical model:

∂2u(x, t)

∂t2− E

ρ

∂2u(x, t)

∂x2= 0 , 0 < x < L, t > 0; (1)

u(0, t) = 0 , M∂2u(L, t)

∂t2+AE

∂u(L, t)

∂x= 0 , t > 0; (2)

u(x, 0) = 0 , 0 ≤ x ≤ L ;∂u(x, 0)

∂t= 0 , 0 ≤ x < L,

∂u(L, 0)

∂t= −α0. (3)

where u(x, t) denotes the displacement of the cross section x of the bar at time t. Here E is the Young’s modulus

of the material of the bar, ρ its constant density and A the area of the uniform cross sections.

The above mathematical model was introduced by Koslyakov et al.[1]

The objective of this paper is to investigate the existence of solutions of Problem (1.1)-(1.3).

2 Main Results

Denote by (u, v) and |u| the usual scalar product and norm of the space L2(0, L) By V is represented the Hilbert

space

V = u ∈ H1(0, L);u(0) = 0

equipped with the scalar product

((u, v)) =

∫ L

0

du

dx

dv

dxdx

and norm ||u|| = ((u, u))1/2. By δL is denoted the fucntional

< δL, v >= v(L) , v ∈ C0([0, L];R) = X

89

90

then δL ∈ X ′.

Let T > 0 be an arbitrary fixed real number. Consider the problem

θ′′(x, t) − θxx(x, t) = f(x, t) , 0 < x < L , 0 < t < T ; (1)

θ(0, t) = 0 , θ′′(L, t) + θx(L, t) = 0 , 0 < t < T ; (2)

θ(x, T ) = 0 , θ′(x, T ) = 0 (3)

where θ′ = ∂θ∂t and θx = ∂θ

∂x . For each f ∈ L1(0, T ;L2(0, L)) is determined the weak solution θ of the Problem

(2,1)-(2.3). One has

1

2|θ′(t)|2 +

1

2||θ(t)||2 +

1

2[θ′(L, t)]2 ≤

∫ T

0

|f(t)||θ′(t)|dt , ∀ 0 ≤ t ≤ T.

Definition 2.1. A function u ∈ L∞(0, T ;L2(0, L)) is named a solution defined by transposition of Problem (1.1)-

(1.3) if ∫ T

0

∫ L

0

u(x, t)f(x, t)dxdt = −2α0 < δL, θ(., 0) > , ∀f ∈ L1(0, T ;L2(0, L))

where θ is the weak solution of (2.1)-(2.3) with f(see [2] and [3])

Theorem 2.1. There exits a unique solution u defined by transposition of Problem (1.1)-(1,3). Furthermore u

satisfies

u′′ − uxx = 0 in L∞(0, T ;L2(0, L));

u(0, .) = 0 , u′′(L, .) + ux(L, .) = 0 in L∞(0, T );

u(0) = 0 , u′(0) = −α0δL.

The theorem is obtained by applying the Galerkin method, results of the Trace Theorem and the interpolation

of Hilbert spaces.

References

[1] Koshlyakov, N.S.; Smirnov, M.M. and Gliner, E.B. -Differential Equations of Mathematical Physics,

North Holland Publishing Company, Amsterdam, 1964

[2] Lions, J.L. and Magenes, E.-Problemes aux Limites Non Homogenes et Applications, Vol. 1, Dunod, Paris,

1968.

[3] Lions, J.L.-Controlabilte Exacte, Perturbations et Stabilisation de Systemes Distribues, Tome 1, Controlabilite

Exacte, Masson, Paris, 1988.


ABOUT POLYNOMIAL STABILITY FOR THE POROUS-ELASTIC SYSTEM WITH FOURIER’S

LAW

ANDERSON RAMOS1, DILBERTO ALMEIDA JUNIOR2 & MIRELSON FREITAS3

1Faculdade de Matematica - Campus de Salinopolis, UFPA, PA, Brasil, [email protected],2Faculdade de Matematica - ICEN, UFPA, PA, Brasil, [email protected],

3Faculdade de Matematica - Campus de Salinopolis, UFPA, PA, Brasil, [email protected]

Abstract

In this work, we consider the porous-elastic equations mixing Kelvin-Voigt dissipation mechanisms and the

thermal effect given by Fourier’s law. We prove that the system lack the exponential decay property for a

particular equality between damping parameters. In that direction, we prove the polynomial decay and the

optimal decay rate.

1 Introduction

Based on Quintanilla and Ueda [1], we consider the one-dimensional porous elastic system with Fourier’s law is

given by

ρutt − αuxx − βϕx − γuxxt − εϕxt + ξθx = 0 in (0, l) × (0,∞), (1)

κϕtt − δϕxx + βux + ηϕ+ τϕt + εuxt − ℏθ = 0 in (0, l) × (0,∞), (2)

ρ2θt −Kθxx + ξuxt + ℏϕt = 0 in (0, l) × (0,∞), (3)

where ρ, α, γ, τ , κ, K, δ, ξ, ℏ and η are constitutive coefficients, β = 0 satisfies αη−β2 > 0. Moreover, the functions

u, ϕ and θ represent the displacement of a solid elastic material, the volume of porous fraction and the temperature,

respectively. In addition, we consider that the coefficients γ > 0, τ > 0 and ε = 0 satisfies the relationship

γτ − ε2 = 0. (4)

Here, we assume the Dirichlet-Neumann-Neumann boundary conditions

u(0, t) = u(l, t) = ϕx(0, t) = ϕx(l, t) = θx(0, t) = θx(l, t) = 0, t > 0, (5)

and the initial conditions

u(x, 0) = u0(x), ut(x, 0) = u1(x), ϕ(x, 0) = ϕ0(x), ϕt(x, 0) = ϕ1(x), θ(x, 0) = θ0(x), x ∈ (0, l). (6)

2 Main Results

Theorem 2.1. Let us suppose that γτ − ε2 = 0. Then the semigroup S(t) = eAt associated with the system (1)–(6)

is not exponentially stable.

91

92

Proof To prove this result we will argue by contradiction, that is, we will show that there exists a sequence of

number(λn)n∈N ⊂ R with |λn| → ∞ and

(Un

)n∈N ⊂ D(A) for

(Fn

)n∈N ⊂ H, with ∥Fn∥H <∞ such that(

iλnI −A)Un = Fn, (1)

where Fn is bounded in H, but ∥Un∥H tends to infinity. To show the lack of exponential stability, we consider(α+ iλγ

)ω2n − λ2ρ

(β + iλε

)ωn −ξωn(

β + iλε)ωn

(η + iλτ

)−(λ2κ− ω2

nδ)

−ℏiλξωn iλℏ iλρ+Kω2

n

An

Bn

Cn

=

0

1

0

. (2)

Solving Eq. (2) we have

Bn ∼iλnγKω

4n +

[K(α− δρ/κ

)− δγρ/κ

]ω4n + O(n3)(

ε2 − τγ)Kδκ ω6

n − iλKγ(w0 − Γ

)ω4n + O(n4)

, (3)

where Γ :=(ηγ − 2εβ

)/γ +

(ε2 − τγ

)δρ/Kγκ+

(α− δρ/κ

)τ/γ. Since γτ − ε2 = 0 and choosing w0 := Γ we have

Bn ∼ O(n). (4)

Therefore,

∥Un∥2H ≥ κ

∫ L

0

|ϕn1 |2dx = κλ2n|Bn|2∫ L

0

| cos(ωnx)|2dx ∼ O(n4) =⇒ limn→∞

∥Un∥2H ≥ κ∥ϕn1∥2 = ∞. (5)

Theorem 2.2 (Polynomial decay). The semigroup associated with the system (1)–(6) satisfies

∥S(t)U0∥H ≤ C

t1/2∥U0∥D(A), ∀ t > 0, U0 ∈ D(A). (6)

Moreover, this rate is optimal.

Proof To show the polynomial stability, we use Borichev and Tomilov’s Theorem [2]. Then using technical lemmas

we get

∥U∥2H ≤ λ2C∥U∥H∥F∥H + C∥U∥H∥F∥H + C∥F∥2H. (7)

Consequently, we have

1

λ2∥(iλI −A)−1F∥H ≤ C∥F∥H, (8)

and therefore, by Borichev and Tomilov result, we prove the polynomial decay.

Now let us suppose that the rate of decay can be improved from t−1/2 to t−1/(2−ϵ) for some ϵ > 0, then we will

have that

1

|λ|2−ϵ∥(iλI −A)−1F∥H, (9)

must be bounded. But this is not possible because of the lack of stability. The proof is now complete.

References

[1] Quintanilla, R. and Ueda, Y. (2020) Decay structures for the equations of porous elasticity in one-

dimensional whole space. Journal of Dynamics and Differential Equations. 32, 1669–1685.

[2] Borichev, A. and Tomilov, Y. (2009) Optimal polynomial decay of functions and Mathematische Annalen.

347, 455–478.


EXISTENCE AND EXPONENTIAL DECAY FOR WAVE EQUATION IN WHOLE HYPERBOLIC

SPACE

P. C. CARRIAO1, O. H. MIYAGAKI2 & A. VICENTE3

1Instituto de Matematica, UFMG, MG, Brasil, [email protected],2Instituto de Matematica, Universidade Federal de Sao Carlos, SP, Brasil, [email protected],

3CCET, Universidade Estadual do Oeste do Parana, PR, Brasil, [email protected]

Abstract

In this work we study the exponential decay of the energy associated to an initial value problem involving the

wave equation on the hyperbolic space BN . The main tools are Faedo-Galerkin method, multipliers techniques,

and an appropriate Hardy inequality.

1 Introduction

In this work we prove the existence of solution and the exponential decay of the energy associated to the following

problem

utt − ∆BNu+ f(u) + a(x)ut = 0 in BN × (0,∞), (1)

u(x, 0) = u0(x), ut(x, 0) = u1(x) for x ∈ BN , (2)

where a, f , u0 and u1 are known functions and ∆BN is the Laplace-Beltrami operator in the disc model of the

Hyperbolic BN . The space BN is the unit disc x ∈ RN : |x| < 1 of RN endowed with the Riemannian metric g

given by gij = p2δij , where p(x) = 21−|x|2 and δij = 1, if i = j and δij = 0, if i = j. The hyperbolic gradient ∇BN

and the hyperbolic Laplacian ∆BN are given by

∇BNu =∇up

and ∆BNu = p−Ndiv(pN−2∇u) = p−2∆ +(N − 2)

px · ∇, (3)

where · is the standard scalar product in RN ; and ∇ and ∆ are the usual gradient and Laplacian of RN .

Since the pioneer work of Zuazua [5], where the author, based on the multiplier techniques and on the unique

continuation results, showed the exponential decay for the semilinear wave equation with localized damping in an

unbounded domains, many authors have been studied this class of problems.

In this work, we extend the work of Zuazua [5] to the hyperbolic space BN which is a non-compact manifold,

with curvature −1 and without boundary. The main idea of our paper is to consider the damping acting away of

the origin, as in [5]. But, now in the context of the hyperbolic space.

The main novelty of our work is to present a new technique which combines the multipliers one with the use

of a Hardy inequality. This techniques was used in the context of elliptic equations in [1, 2, 3], but in evolution

problem it is a novelty.

References

[1] carriao, p. c., costa, a. c. r., and miyagaki, o. h. - A class of critical Kirchhoff problem on the hyperbolic

space Hn, Glasgow Math. J., doi:10.1017/S0017089518000563, 2019.

93

94

[2] carriao, p. c., costa, a. c. r., miyagaki, o. h., and vicente a. - Kirchhoff-type problems with critical

Sobolev exponent in a hyperbolic space, Electronic Journal of Differential Equations, Vol. 2021, 53, 1-12 2021.

[3] carriao, p. c., lehrer, r., miyagaki, o. h., and vicente a. - A Brezis-Nirenberg problem on hyperbolic

spaces, Electronic Journal of Differential Equations, Vol. 2019, 67, 1-15, 2019.

[4] carriao, p. c., miyagaki, o. h., vicente, a., Exponential decay for semilinear wave equation with localized

damping in the hyperbolic space, Mathematische Nachrichten, to appear.

[5] zuazua, e. - Exponential decay for the semilinear wave equation with localized damping in unbounded domains,

J. Math. pures et appl. 70, 513-529, 1991.


REALIZABILITY OF THE RAPID DISTORTION THEORY SPECTRUM

AILIN RUIZ DE ZARATE FABREGAS1, NELSON LUIS DIAS2 & DANIEL G. ALFARO VIGO3

1Department of Mathematics, UFPR, PR, Brazil, [email protected],2Department of Environmental Engineering, UFPR, PR, Brazil, [email protected],

3Department of Computer Science, Institute of Mathematics, UFRJ, RJ, Brazil, [email protected]

Abstract

In this work we show that the Rapid Distortion Theory (RDT) model for the spectral tensor of the

homogeneous turbulence problem in the whole three-dimensional domain preserves the symmetry, positive

semidefiniteness and integrability properties required in Cramer’s characterization of the spectral tensor of a

continuous homogeneous random process. The correlation tensor recovered from the spectral tensor model is

statistically valid and satisfies realizability conditions. The RDT spectral tensor model is a system of transport

equations plus an algebraic restriction due to incompressibility, therefore, we deal with the existence, uniqueness

and persistence of solutions in a specific set of functions by using DiPerna-Lions renormalization techniques.

1 Introduction

We consider an incompressible fluid in the whole three-dimensional domain R3 with constant density ρ, kinematic

viscosity ν and in constant-shear flow. Starting from the equation of continuity and the Navier-Stokes equations in

the continuous homogeneous random process framework known as homogeneous turbulence, it is possible to derive

the equations for the evolution of the spectral tensor Φ(k, t), where k denotes the wavenumber vector and t is the

temporal variable. The spectral tensor is the spatial Fourier transform of the velocity correlation tensor R which is

defined componentwise as Rij(r, t) = ⟨ui(x, t)uj(x + r, t)⟩, (r, t) ∈ R3×R, t ≥ 0, where ui(x, t) = Ui(x, t)−⟨Ui(x, t)⟩denotes the fluctuations of each component of the random velocity field Ui, i = 1, 2, 3, and ⟨·⟩ is the notation for

the expected value. Note that R and any statistical moment of order n ≥ 2 are invariant under spatial translations

because of the homogeneous turbulence assumption.

The evolution equations for the spectral tensor are part of an infinite hierarchy of coupled nonlinear integro-

differential equations such that the equations for the statistical moments of order n involve the moments of order

n + 1. That is why a closure scheme is introduced, the simplest of which consists of discarding the higher-order

moments terms in the equations for the highest-order moments considered, as it is the case of the RDT model in

which third-order moments are discarded. As a result, the following first-order homogeneous linear system arises

∂ΦRij

∂t= Cnm

[km

∂ΦRij

∂kn+

2kmk2

(kiΦ

Rnj + kjΦ

Rin

)−(δimΦR

nj + δjmΦRin

)]− 2νk2ΦR

ij , (1)

where summation convention is adopted for repeated Latin indices, the superscript R indicates that ΦRij is in general

different from Φij and a constant mean velocity gradient

Cnm =∂ ⟨Um⟩∂xn

= δn3δm1C31, C31 = 0, (2)

is assumed, which corresponds to a constant shear rate of the mean velocity ⟨U1⟩ in the x3 direction. Besides, the

continuity equation expressed in the form of the incompressibility condition in the physical domain returns

ΦRijkj = 0, (3)

95

96

adding an algebraic equation to be satisfied by ΦR. The approximation for Φ provided by the RDT spectral tensor

model has been validated in practice for rapidly straining turbulent flow, as described in [1]. It is then pertinent

to analyze if the model preserves the statistical properties that the spectral tensor of a continuous homogeneous

random processes must satisfy, as considered in the next section.

2 Main Results

Cramer’s theorem provides a characterization of the spectral tensor of a continuous homogeneous random process

as a k-absolutely integrable Hermitian matrix representing a positive semidefinite quadratic form. Simplifying to

the real valued case, we define an admissible initial condition as a real absolutely integrable symmetric positive

semidefinite matrix Φ0(k) such that Φ0(k)k = 0 almost everywhere, which serves as initial condition for system (1)–

(3). In this setting, the following theorem establishes the fulfillment of Cramer’s characterization together with the

existence and uniqueness of solutions for the model.

Theorem 2.1. If Φ0 is an admissible initial condition then there exists a matrix ΦR such that

1. ΦR is symmetric and positive semidefinite,

2. ΦR(k, t)k = 0, for all t ≥ 0 and almost everywhere in k,

3. the components of ΦR are continuous functions with respect to t, for t ≥ 0, with values in L1(R3),

4. ΦR is the unique weak solution of the system of equations (1), with initial condition Φ0, that satisfies property 3

5. If Φ0 is also a continuously differentiable function on R3−0 then ΦR is also continuously differentiable on

(k, t) ∈ R3 − 0 × R, t ≥ 0.

A detailed proof can be found in [2]. It relies on the use of the Kelvin-Townsend system of differential

equations which solutions serve as factors for the complete solution. The structure of the system that allows

this factorization approach is intrinsic to the original equations. The results are also valid for the complex

Hermitian case. A statistically valid correlation tensor is recovered from the spectral tensor model via inverse

Fourier transform. Therefore, it satisfies physically meaningful probabilistic inequalities, including, at zero spatial

separation, realizability conditions for the Reynolds stress tensor R(0, t).

Acknowledgments

A. Ruiz de Zarate Fabregas is grateful to the Graduate Program in Environmental Engineering and the Mathematics

Department at the Federal University of Parana (UFPR) for making possible her one-year sabbatical leave at the

Laboratory for Environmental Monitoring and Modeling Analysis (Lemma), UFPR.

N. L. Dias’ work has been partially supported by Brazil’s CNPq research grant 301420/2017-3.

References

[1] hunt, j. c. r. and carruthers, d. j. - Rapid distortion theory and the ‘problems’ of turbulence. Journal

of Fluid Mechanics, 212, 497-537, 1990.

[2] ruiz de zarate fabregas, a., dias, n. l. and alfaro vigo d. g. - Realizability of the rapid distortion

theory spectrum: the mechanism behind the Kelvin-Townsend equations. Journal of Mathematical Physics,

62, 063101, 2021.


STABILITY OF PERIODIC SOLUTIONS OF THE NAVIER-STOKES EQUATIONS

ENRIQUE FERNANDEZ-CARA1, FELIPE WERGETE CRUZ2 & MARKO A. ROJAS-MEDAR3

1EDAN-IMUS, Universidad de Sevilla, Spain, [email protected],2Departamento de Matematica, Universidade Federal de Pernambuco, Recife-PE, Brazil, [email protected],

3Departamento de Matematica, Universidad de Tarapaca, Arica, Chile, [email protected]

Abstract

We establish the existence of periodic solutions for the Navier-Stokes equations, assuming that the external

force is periodic and C1 in time, and small enough in the norm of the considered space. We also prove uniqueness

and stability of the solutions in various norms. The proof of existence is based on a set of estimates for the

family of finite-dimensional approximate solutions.

1 Introduction

1.1 Problem Statement

Given a periodic force in time f : Ω × R → Rn, Ω ⊂ Rn, n = 2 or 3, f(x, t) = f(x, t+ τ), we search for a periodic

solution in time

u : Ω × R → Rn u(x, t) = u(x, t+ τ)

p : Ω × R → R p(x, t) = p(x, t+ τ)

of the Navier-Stokes equations ∂u

∂t+ (u · ∇)u − µ∆u + ∇p = f ,

div u = 0,(1)

subject to the following boundary condition:

u(x, t) = 0 on ∂Ω × (0, T ). (2)

Also, we consider the initial-value boundary problem associated to (1)–(2), i.e. (1)–(2) together with

u(x, 0) = u0(x) in Ω . (3)

1.2 Preliminaries

We use the usual function spaces for the Navier-Stokes equations, see Lions [1]. We will denote by ∥ · ∥ the usual

norm in L2(Ω) and associated product spaces.

The following results on existence and uniqueness can be found for instance in [6, 3, 5]:

Theorem 1.1. Let f ∈ C1(τ ;L2(Ω)). There exists M1 > 0 such that, if

supt

∥f(t)∥ ≤M1, (4)

the corresponding system (1)–(2) has a strong τ−periodic solution

up ∈ H2(τ ;H) ∩H1(τ ;D(A)) ∩ L∞(τ ;D(A)) ∩W 1,∞(τ ;V ). (5)

97

98

Moreover, if ∥f∥C1(τ ;L2(Ω)) is small enough, the solution is unique and any solution u to (1)–(3) defined for all

t ∈ (0,+∞) with values in D(A) and such that

sup0≤t≤T

∥u(t)∥ +

∫ T

0

∥∇u(t)∥2dt < +∞ ∀T > 0 (6)

satisfies ∥u(t) − up(t)∥ → 0 exponentially as t→ +∞.

2 Main Results

Theorem 2.1. If f and u are as in the second part of Theorem 2.3, one has

∥u(t) − up(t)∥H2 → 0 exponentially as t→ +∞. (1)

The proof can be achieved by establishing appropriate estimates of ∥u− up∥ in some norms.

Theorem 2.2. If f and u are as in the second part of Theorem 2.3, ∂Ω ∈ C∞, Dltf ∈ L∞(0,∞;Hk(Ω)) for all

l, k ≥ 0 and u0 ∈ D(A), we have

∥Dnt u(t) −Dn

t up(t)∥Hk → 0 exponentially as t→ +∞. (2)

The proof can be obtained by induction, following some arguments already used in [2] and [1] together with the

arguments in the proof of Theorem 2.1.

Acknowledgment: Partially supported by: MATH-AMSUD project 21-MATH-03 (CTMicrAAPDEs),

CAPES-PRINT 88887.311962/2018-00 (Brazil), Project UTA-Mayor, 4753-20, Universidad de Tarapaca (Chile),

CONICYT- PCI/ Atraccion de Capital Humano Avanzado del Extranjero Nro. 80170081 (Chile)

References

[1] guermond, j-l. Faedo-Galerkin weak solutions of the Navier-Stokes equations with Dirichlet boundary

conditions are suitable. J. Math. Pures Appl. (9) 88, 87-106, 2007.

[2] heywood, j. g., rannacher, rolf- Finite element approximation of the nonstationary Navier-Stokes

problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer.

Anal. 19, 275-311, 1982.

[3] kato, h.- Existence of periodic solutions of the Navier-Stokes equations. J. Math. Anal. Appl. 208, 141-157,

1997.

[4] lions, j. l. - Quelques methodes de resolution des problemes aux limites non lineares., Dunod-Gauthier Villars,

Paris, First edition, 1969.

[5] lukaszewicz, g., ortega-torres, e., rojas-medar, m.a.- Strong periodic solutions for a class of abstract

evolution equations. Nonlinear Anal. 54, 1045-1056, 2003.

[6] serrin, j.- A note on the existence of periodic solutions of the Navier-Stokes equations. Arch. Rational Mech.

Anal. 3, 120-122, 1959.


A DAMPED NONLINEAR HYPERBOLIC EQUATION WITH NONLINEAR STRAIN TERM

EUGENIO CABANILLAS L.1, ZACARIAS HUARINGA S.2, JUAN B. BERNUI B.3 & BENIGNO GODOY T.4

1Instituto de Investigacion, Facultad de Ciencias Matematicas-UNMSM, Lima-Peru, [email protected],2Instituto de Investigacion, Facultad de Ciencias Matematicas-UNMSM, Lima-Peru, [email protected],

3 Instituto de Investigacion, Facultad de Ciencias Matematicas-UNMSM, Lima-Peru, [email protected],4 Instituto de Investigacion, Facultad de Ciencias Matematicas-UNMSM, Lima-Peru, [email protected]

Abstract

In this work, we investigate an initial boundary value problem related to the nonlinear hyperbolic equation

utt + uxxxx + αuxxxxt = f(ux)x, for f(s) = |s|ρ + |s|σ, 1 < ρ, σ, α > 0. Under suitable conditions, we prove the

existence of global solutions and the exponential decay of energy.

1 Introduction

In this research, we consider the initial boundary value problem of a nonlinear hyperbolic equation with Kelvin-Voigt

type damping term

utt + uxxxx + αuxxxxt = f(ux)x, x ∈ Ω, t > 0,

u(0, t) = u(1, t) = 0, uxx(0, t) = uxx(1, t) = 0, t ≥ 0, (1)

u(x, 0) = u0(x), ut(x, 0) = u1(x),

where Ω = (0, 1), α is a positive constant and f(s) = |s|ρ + |s|σ is the strain term.

There have been many impressive works on the well-posedness and energy decay of solutions of the nonlinear

beam equations of the type (1), with f(s) satisfying f(s)s ≥ 0 and |f(s)| ≤ a|s|q, a > 0, see [1] and the references

therein. The authors in these works introduced the potential well method to obtain their results.

However, to the best of our knowledge, there is little research about equation (1) with nonlinearity f(s) =

|s|ρ + |s|σ, ρ, σ > 1, which also implies that (1) has no positive definite energy.

2 Main Results

We define the space

H = u ∈ H3(Ω) ∩H10 (Ω) : uxx ∈ H1

0 (Ω)

endowed with the norm

∥u∥2H = ∥uxx∥2 + ∥uxxx∥2,

and introduce the following notations

B1 = supu∈H2(Ω)∩H1

0 (Ω)u=0

∥ux∥ρ+1

∥uxx∥, B2 = sup

u∈H2(Ω)∩H10 (Ω)

u =0

∥ux∥σ+1

∥uxx∥,

γ1 =1

ρ+ 1Bρ+1

1 , γ2 =1

σ + 1Bσ+1

2 .

99

100

Let λ1 = min[

14(ρ+1)γ1

]1/(ρ−1)

,[

16(σ+1)γ2

]1/(σ−1) and the energy associated with problem (1) by

E(t) =1

2∥ut∥2 +

1

2∥uxx∥2 −

1

ρ+ 1

∫ 1

0

|ux|ρux dx− 1

σ + 1

∫ 1

0

|ux|σux dx.

Now, we have the following two theorems

Theorem 2.1. Assume u0 ∈ H, u1 ∈ L2(Ω). Then problem (1) has a unique weak solution u for T small enough.

Proof. We prove the local existence of weak solutions by using the Faedo-Galerkin method.

Theorem 2.2 (The main result.). Assume that the assumptions of Theorem 2.1 hold and that

0 < ∥u0xx∥ < λ1 and (4E(0))1/2

< λ1,

where

E(0) =1

2∥u1∥2 +

1

2∥u0xx∥2 −

1

ρ+ 1

∫Ω

|u0x|ρu0x dx− 1

σ + 1

∫Ω

|u0x|σu0x dx.

Then problem admits a global weak solution in time. This solution satisfies

E(t) ≤ L0e−γt,∀t ≥ 0, for some L0, γ > 0.

Proof. We apply an argument due to Tartar [4] (See also Milla Miranda et al.[2]) to get the global existence. The

exponential decay is obtained via an integral inequality introduced by Komornik.

References

[1] Lian W., Xu R.Z., Radulescu V.D., Yang Y.B., Zhao N. - Global well-posedness for a class of fourth

order nonlinear strongly damped wave equations, Adv. Calc. Var., http://dx.doi.org/10.1515/acv-2019-003,

2019.

[2] Milla Miranda M., Louredo A.T. Clark M.R., Clark H.R. - On energy wave equations with non-

negative non-linear terms. Int. J. Non Linear Mech., 82, 6-16, 2016.

[3] Pang T., Shen J. - Global existence and asymptotic behaviour of solution for a damped nonlinear hyperbolic

equation, Nonlinear Anal., 198, Article 111885, 2020.

[4] Tartar L. - Topics in Nonlinear Analysis, Publications Mathematiques d’Orsay, Uni. Paris Sud. Dep.

Math., Orsay, France, 1978.


COMPORTAMENTO ASSINTOTICO PARA AS EQUACOES MAGNETO-MICROPOLARES

FELIPE W. CRUZ1, CILON PERUSATO2, MARKO ROJAS-MEDAR3 & PAULO ZINGANO4

1Departamento de Matematica, UFPE, PE, Brasil, [email protected],2Departamento de Matematica, UFPE, PE, Brasil, [email protected],

3Departamento de Matematica, Universidad de Tarapaca, Chile, [email protected],4 Departamento de Matematica Pura e Aplicada, UFRGS, RS, Brasil, [email protected]

Abstract

Estudamos o comportamento assintotico das solucoes globais fracas para as equacoes dos fluidos magneto-

micropolares nos espacos de Sobolev Hm(Rn), com m ∈ N ∪ 0 e n ∈ 2, 3. Alem disso, mostramos que a

velocidade micro-rotacional decai mais rapido do que a velocidade linear do fluido. Tambem discutimos alguns

resultados de decaimento para a pressao total do fluido e para as derivadas da solucao na regiao do espaco-tempo.

1 Introducao

Consideramos o PVI

ut + (u · ∇)u− (µ+ χ)∆u + ∇(Π +

|b|2

2

)= (b · ∇)b + χ rotw,

wt + (u · ∇)w − γ∆w − κ∇(divw) + 2χw = χ rotu,

bt + (u · ∇)b− ν∆b = (b · ∇)u,

divu = div b = 0,

(u,w, b)|t=0 = (u0,w0, b0) ,

(1)

em Rn × (0,∞), onde u0, w0 e b0 sao funcoes dadas e n = 2 ou 3.

No sistema (1), as incognitas sao as funcoes u(x, t) ∈ Rn, Π (x, t) ∈ R, w(x, t) ∈ Rn e b(x, t) ∈ Rn, as quais

representam, respectivamente, o campo velocidade incompressAvel (velocidade linear), a pressao hidrostatica, a

velocidade micro-rotacional e o campo magnetico em um ponto x ∈ Rn no tempo t > 0. A funcao |b|2/2 e a

pressao magnetica. Assim, denotamos por p := Π + |b|2/2 a pressao total do fluido. Este sistema descreve o

movimento de um fluido incompressAvel micropolar viscoso na presenca de um campo magnetico (veja [1] e [2]).

As constantes positivas µ, χ, γ, κ e ν estao associadas a propriedades especAficas do fluido; mais especificamente, µ

e a viscosidade cinematica (usual), χ e a viscosidade do vortice, γ e κ sao as viscosidades de rotacao e, por ultimo,

1/ν e o numero magnetico de Reynolds. Os dados iniciais para os campos velocidade e magnetico, dados por u0

e b0, sao assumidos livres de divergente, i.e., divu0 = div b0 = 0. Vale ressaltar que o sistema (1) se reduz A s

equacoes de Navier-Stokes, quando w = b = 0; ao sistema MHD, quando w = 0; e ao sistema micropolar, quando

b = 0.

101

102


Por simplicidade, assumimos µ = χ = 1/2 e γ = κ = ν = 1.

Teorema 2.1. Seja (u, p,w, b) uma solucao global do sistema (1). Se u0,w0, b0 ∈ Hm(Rn) ∩ L1(Rn), com

divu0 = div b0 = 0, e m ∈ N ∪ 0 e n ∈ 2, 3, entao∥∥Dmu(·, t)∥∥L2(Rn)

+∥∥Dmw(·, t)

∥∥L2(Rn)

+∥∥Dm b(·, t)

∥∥L2(Rn)

≤ C(t+ 1)−m2 −n

4 , (1a)

para todo t suficientemente grande. Ademais, temos a seguinte taxa de decaimento melhorada para a micro-rotacao:∥∥Dmw(·, t)∥∥L2(Rn)

≤ C (t+ 1)−m2 − n

4 − 12 , ∀ t≫ 1. (1b)

Tambem comparamos a evolucao das solucoes z(·, t) := (u,w, b)(·, t) do sistema (1) com as solucoes z(·, t) :=

(u,w, b)(·, t) do sistema linear associado. Em [3], M. Wiegner forneceu tal estimativa para as equacoes de Navier-

Stokes (comparando com a equacao do calor com os mesmos dados iniciais). Embora tenhamos outro sistema linear

associado, o resultado permanece valido e, para o campo micro-rotacional, fornecemos uma taxa de decaimento

extra. Nosso segundo resultado principal e o seguinte

Teorema 2.2. Seja (u, p,w, b) uma solucao global do sistema (1). Se u0,w0, b0 ∈ L1(Rn) ∩ H1(Rn), com

divu0 = div b0 = 0, entao existe uma constante C ∈ R+ tal que

∥z(·, t) − z(·, t)∥L2(Rn) ≤ C (t+ 1)−n4 − 1

2 , ∀ t ≥ 0. (2a)

Alem disso, melhoramos a taxa de decaimento para o campo micro-rotacional da seguinte forma:

∥w(·, t) −w(·, t)∥L2(Rn) ≤ C (t+ 1)−n4 −1, ∀ t ≥ 0. (2b)

Observacao 1. Note que o resultado acima nos diz que as solucoes do sistema magneto-micropolar sao

assintoticamente equivalente A s solucoes do problema linear associado com os mesmos dados iniciais.

Por fim, para u0,w0, b0 ∈ L1(Rn) ∩Hm+1(Rn), com divu0 = div b0 = 0, tambem obtivemos a seguinte taxa

de decaimento para a pressao total do fluido∥∥Dm p(·, t)∥∥L2(Rn)

≤ C (t+ 1)−m2 − 3n

4 , ∀ t≫ 1,

e, supondo que u0,w0, b0 ∈ L1(Rn) ∩ HM (Rn), com divu0 = div b0 = 0, tambem mostramos que existe uma

constante C > 0 tal que ∥∥Dm ∂kt u(·, t)∥∥L2(Rn)

≤ C(t+ 1)−m2 −k−n

4 , ∀ t≫ 1,∥∥Dm ∂kt w(·, t)∥∥L2(Rn)

≤ C(t+ 1)−m2 −k−n

4 − 12 , ∀ t≫ 1,∥∥Dm ∂kt b(·, t)

∥∥L2(Rn)

≤ C(t+ 1)−m2 −k−n

4 , ∀ t≫ 1,

para todo M ≥ m+ 2k, m, k ∈ N ∪ 0 e n ∈ 2, 3.

References

[1] lukaszewicz, g. - Micropolar Fluids: Theory and Applications. Model. Simul. Sci. Eng. Technol., Birkhauser,

Boston, 1999.

[2] rojas-medar, m. a. - Magneto-micropolar fluid motion: Existence and uniqueness of strong solution. Math.

Nachr. 188, 301–319, 1997.

[3] wiegner, m. - Decay results for weak solutions of the Navier-Stokes equations on Rn. J. London Math. Soc.

35, 303–313, 1987.


RESULTADOS DE EXISTENCIA GLOBAL PARA SOLUCOES DE EQUACOES DE

ADVECCAO-DIFUSAO

JANAINA P. ZINGANO1, JULIANA S. ZIEBELL2, LINEIA SCHUTZ3 & PATRICIA L. GUIDOLIN4

1Instituto de Matematica, UFRGS, RS, Brasil, [email protected],2Instituto de Matematica, UFRGS, RS, Brasil, [email protected],3Instituto de Matematica, UFRGS, RS, Brasil, [email protected],

4 Instituto de Matematica, UFRGS, RS, Brasil, [email protected]

Abstract

Neste trabalho usamos uma tecnica baseada em metodos de energia para analisar a existencia global da

solucao do problema evolutivo ut + (b(x, t)uk+1)x = µ(t)uxx com condicao inicial u(·, 0) = u0 ∈ L1(R)∩L∞(R).Encontramos condicoes que garantam a existencia global da solucao.

1 Introducao

Neste trabalho, apresentamos um estudo detalhado sobre o comportamento assintotico de solucoes limitadas nao

negativas do problema evolutivo do tipo

ut + (b(x, t)uk+1)x = µ(t)uxx ∀x ∈ R, t > 0,

u(·, 0) = u0 ∈ L1(R) ∩ L∞(R), (1)

para campos de adveccao arbitrarios continuamente diferenciaveis b satisfazendo

b(·, t) ∈ L∞loc([0,∞), L∞(R)) ∀x ∈ R, t ≥ 0; (2)∣∣∣∣∂b(x, t)∂x

∣∣∣∣ < B(t) ∀x ∈ R, t ≥ 0, (3)

onde B ∈ C0([0,∞)), µ(t) ∈ C0([0,∞)) positiva e 0 ≤ k < 2 constante.

Aqui, por uma solucao limitada do problema (1) em algum intervalo de tempo [0, T∗), entendemos como qualquer

funcao u ∈ C0([0, T∗), L1(R)) ∩ L∞loc([0, T∗), L∞(R)) satisfazendo a equacao do problema (1). Para resultados de

existencia local (no tempo) pode-se consultar [2] and [6], Ch. 7.

Nosso objetivo e investigar para quais valores de k podemos garantir que a solucao do problema exista

globalmente e por este motivo e tao importante conhecer o comportamento das normas mais altas Lq, em especial

da norma L∞ no intervalo de existencia da solucao, a fim de que possamos estender este intervalo a intervalos de

existencia mais amplos.

Note que para k = 1 em (1), temos a equacao de Burgers com um termo advectivo arbitrario b(x, t), apesar do

vasto estudo a respeito desta equacao, um dos modelos mais simples que combina os efeitos do operador nao linear

advectivo com o operador difusivo, a dependencia explicita de x em b dificulta bastante a analise da existencia

global das solucoes deste problema. Para ilustrar esta dificuldade, reescrevemos a equacao geral do problema (1)

da seguinte forma

ut + (k + 1)b(x, t)ukux = −∂b(x, t)∂x

uk+1 + µ(t)uxx. (4)

103

104

Como b depende de x, na regiao onde∂b(x, t)

∂x< 0, o termo −∂b(x, t)

∂xuk+1 estimula a solucao u a crescer em

magnitude. Porem, a solucao de (1) conserva massa, entao a medida que u cresce, o perfil da solucao fica mais

afinado, tornando-se mais suscetıvel aos efeitos do operador difusivo. Desta forma, o resultado da competicao entre

os termos difusivo e advectivo do lado esquerdo da equacao (4) torna-se difıcil de ser previsto.

Quando b nao depende explicitamente de x, ou mais geralmente, quando ∂b(x, t)/∂x ≥ 0 for all x ∈ R, ja se

sabe da existencia global para as solucoes no caso k = 0 em (1), e neste caso, alem das solucoes serem definidas

para todo tempo, elas tambem decaem a zero quando t −→ ∞, ver os seguintes trabalhos nesta direcao [4, 1, 7].

Para o caso de k = 0 e b(x, t) arbitraria, ver [3, 5]. Finalmente, no nosso trabalho, para b(x, t) arbitraria e k ∈ [0, 2)

garantimos a existencia global.


Obtemos uma estimativa para a norma do sup da solucao do problema (1):

Teorema 2.1. Let q ≥ 1, 0 ≤ t0 < t < T∗ and 0 ≤ k < 2 in (1). Then

∥u(·, t)∥L∞(R) ≤ q1

2q−k max∥u(·, t0)∥L∞(R);Bµ(t0; t)

12q−kUq(t0; t)

2q2q−k

,

where Uq(t0; t), Bµ(t0; t) is defined by

Uq(t0; t) := sup∥u(·, τ)∥Lq(R); t0 ≤ τ ≤ t, and Bµ(t0; t) := sup

B(τ)

µ(τ); t0 ≤ τ ≤ t

, respectivamente.

Em particular, se tomarmos t0 = 0 e q = 1 no Teorema 2.1, sabendo que a solucao do problema (1) conserva

massa, garantimos a sua existencia global, isto e, T∗ = ∞.

References

[1] Escobedo, M. and Zuazua, E., Large time behavior for convection-diffusion equations in Rn, Journal of

Functional Analysis, vol. 100, no. 1, pp. 119?161, 1991.

[2] Ladyzhenskaya,O. A., Solonnikov, V. A. and Uralceva, N. N., Linear and Quasilinear Equations of Parabolic

Type, American Mathematical Society, Providence, 1968.

[3] Braz e Silva, P., Melo, W. G., and Zingano, P. R., An asymptotic supnorm estimate for solutions of 1-D

systems of convection-diffusion equations, J. Diff. Eqs. 258, 2806-2822, (2015).

[4] Braz e Silva, P., SchA¼tz, L., and Zingano, P. R.,On some energy inequalities and supnorm estimates for

advection-diffusion equations in Rn, Nonlinear Analysis:Theory, Methods and Applications, vol. 93, pp. 90?96,

2013.

[5] Barrionuevo, J. A., Oliveira, L. S., and Zingano, P. R., General asymptotic supnorm estimates for solutions

of one-dimensional advection-diffusion equations in heterogeneous media, Intern. J. Partial Diff. Equations

(2014), 1-8

[6] Serre, D. Systems of Conservation Laws, vol. 1, Cambridge University Press, Cambridge, 1999.

[7] Schonbek, M. E.,Uniform decay rates for parabolic conservation laws, Nonlinear Analysis, vol. 10, no. 9, pp.

943?956, 1986.


LONG-TIME DYNAMICS FOR A FRACTIONAL PIEZOELECTRIC SYSTEM WITH MAGNETIC

EFFECTS AND FOURIER’S LAW

MIRELSON FREITAS1, ANDERSON RAMOS2, DILBERTO ALMEIDA JUNIOR3 & AHMET OZER4

1Faculdade de Matematica, UFPA, Salinopolis–PA, Brasil, [email protected],2Faculdade de Matematica, UFPA, Salinopolis–PA, Brasil, [email protected],3Faculdade de Matematica, UFPA, Belem–PA, Brasil, [email protected],

4 Department of Mathematics, WKU, Bowling Green, USA, [email protected]

Abstract

In this work, we use a variational approach to model vibrations on a piezoelectric beam with fractional

damping depending on a parameter ν ∈ (0, 1/2). Magnetic and thermal effects are taken into account via the

Maxwell’s equations and Fourier law, respectively. Existence and uniqueness of solutions of the system is proved

by the semigroup theory. The existence of smooth global attractors with finite fractal dimension and the existence

of exponential attractors for the associated dynamical system are proved. Finally, the upper-semicontinuity of

global attractors as ν → 0+ is shown.

1 Introduction

In this work, we consider the longitudinal vibrations on a piezoelectric beam system with thermal and magnetic

effects and with friction dampingρvtt − αvxx + γβpxx + δθx + f1(v, p) = h1 in (0, L) × (0, T ),

µptt − βpxx + γβvxx +Aνpt + f2(v, p) = h2 in (0, L) × (0, T ),

cθt − κθxx + δvxt = 0 in (0, L) × (0, T ),

(1)

where the physical constants ρ, α, β, γ, δ, κ, µ and c are positive constants, f1, f2 are nonlinear source terms

and h1, h2 are external forces. Moreover, we consider the relationship α = α1 + γ2β with α1 > 0. Moreover,

A : D(A) ⊂ L2(0, L) → L2(0, L) is the one-dimensional Laplacian operator defined by

A = −∂xx, with domain D(A) =v ∈ H2(0, L) ∩H1

∗ (0, L) : vx(L) = 0

(2)

where H1∗ (0, L) :=

u ∈ H1(0, L); u(0) = 0

and Aν : D(Aν) ⊂ L2(0, L) → L2(0, L) is the fractional power

associated with operator A of order ν ∈ (0, 1/2).

The system (Pj) is supplemented by the clamped-free boundary and initial conditions

v(0, t) = αvx(L, t) − γβpx(L, t) = 0, t > 0,

p(0, t) = px(L, t) − γvx(L, t) = 0, t > 0,

θ(0, t) = θ(L, t) = 0, t > 0,

v(x, 0) = v0, vt(x, 0) = v1, 0 < x < L,

p(x, 0) = p0, pt(x, 0) = p1, 0 < x < L,

θ(x, 0) = θ0(x), 0 < x < L.

(3)

We assume that

105

106

(i) The external forces h1, h2 ∈ L2(0, L);

(ii) There exists a function F ∈ C2(R2) such that

∇F = (f1, f2); (4)

(iii) There exist q ≥ 1 and C > 0 such that

|∇fj(v, p)| ≤ C(|v|q−1 + |p|q−1 + 1

), j = 1, 2; (5)

(iv) There exist constants η ≥ 0, mF > 0 such that

F (v, p) ≥ −η(|v|2 + |p|2

)−mF , ∇F (v, p) · (v, p) − F (v, p) ≥ −η

(|v|2 + |p|2

)−mF . (6)

2 Main Results

First, we observe that the system (Pj)-(3) defines a dynamical system (H, S(t)). To this system we study the

existence of global attractors and their properties.

Theorem 2.1. Suppose that assumptions (4)-(6) hold. Then,

(i) The dynamical system (H, S(t)) possesses a unique compact global attractor A ⊂ H;

(ii) The global attractor A has finite fractal and Hausdorff dimension;

(iii) The complete trajectories (v, p, vt, pt, θ) in A has further regularity

∥(v, p)∥2(H2(0,L)∩H1∗(0,L))2 + ∥θ∥H2(0,L)∩H1

0 (0,L) + ∥(vt, pt)∥2(H1∗(0,L))2

+∥(vtt, ptt)∥2(L2(0,L))2 + ∥θt∥22 ≤ C, (1)

for some constant C > 0;

(iv) The dynamical system (H, S(t)) has a generalized exponential attractor Aexp with finite fractal dimension in

the extended space

H−1 := (L2(0, L))2 × (H−1∗ (0, L))2 ×H−1(0, L), (2)

where H−1∗ (0, L) is the dual space of H1

∗ (0, L) pivoted with respect to L2(0, L). In addition, from interpolation

theorem, there exists a generalized exponential attractor whose fractal dimension is finite in a smaller extended

space H−σ, where

H = H0 ⊂ H−σ ⊂ H−1, 0 < σ ≤ 1. (3)

References

[1] Chueshov, I. - Dynamics of Quasi-Stable Dissipative Systems, Springer, Berlin 2015.

[2] Chueshov, I. and Lasiecka, I. - Von Karman Evolution Equations. Well-posedness and Long Time

Dynamics, Springer Monographs in Mathematics, New York, 2010.

[3] Freitas, M. M., Ramos, A. J. A., Ozer, A. O. and Almeida Junior, D. S. - Long-time dynamics for a

fractional piezoelectric system with magnetic effects and Fourier’s law. Journal of Differential Equations, 280,

891-927, 2021.


GLOBAL SOLUTIONS TO THE NON-LOCAL NAVIER-STOKES EQUATIONS

JOELMA AZEVEDO1, JUAN CARLOS POZO2 & ARLUCIO VIANA3

1Universidade de Pernambuco, UPE, PE, Brasil, [email protected],2Facultad de Ciencias, Universidad de Chile, Santiago, Chile, [email protected],

3Universidade Federal de Sergipe, UFS, SE, Brasil, [email protected]

Abstract

We study the global well-posedness for a non-local-in-time Navier-Stokes equation. Our results recover in

particular other existing well-posedness results for the Navier-Stokes equations and their time-fractional version.

We show the appropriate manner to apply Kato’s strategy and this context, with initial conditions in the

divergence-free Lebesgue space Lσd (Rd).

1 Introduction

Consider the fractional-in-time Navier-Stokes equation

∂αt u− ∆u+ (u · ∇)u+ ∇p = f, t > 0, x ∈ Ω ⊂ Rd,

∇ · u = 0, t > 0, x ∈ Ω ⊂ Rd,

u(0, x) = u0(x), x ∈ Ω ⊂ Rd,

where ∂αt u denotes the fractional derivative of u in the Caputo’s sense with order α ∈ (0, 1). If the product

(k ∗ v) denotes the convolution on the positive halfline R+ := [0,∞) with respect to time variable, then we have

∂αt u = g1−α ∗ ut, for an absolutely continuous function u, where gβ is the standard notation for the function

gβ(t) = tβ−1

Γ(β) , t > 0, β > 0. Toward the possibility of considering more general nonlocal-in-time effects, we will

replace gα by k, and we assume as a general hypothesis that k is a kernel of type (PC), by which we mean that the

following condition is satisfied:

(PC) k ∈ L1,loc(R+) is nonnegative and nonincreasing, and there exists a kernel ℓ ∈ L1,loc(R+) such that k ∗ ℓ = 1

on (0,∞).

We also write (k, ℓ) ∈ PC. We point out that the kernels of type (PC) are called Sonine kernels and they have been

successfully used to study integral equations of first kind in the spaces of Holder continuous, Lebesgue and Sobolev

functions, see [1].

Therefore, we consider the following problem for the following nonlocal-in-time Navier-Stoke-type equation

∂t(k ∗ (u− u0)) − ∆u+ (u · ∇)u+ ∇p = f, t > 0, x ∈ Rd, (1)

∇ · u = 0, t > 0, x ∈ Rd, (2)

u(0, x) = u0(x), x ∈ Rd, (3)

where u(t, x) represents the velocity field and p(t, x) is the associated pressure of the fluid. The function

u0(x) = u(0, x) is the initial velocity and f(t, x) represents an external force. The problem (1)-(3), can be written

in an abstract form as

∂t(k ∗ (u− u0)) + Apu = F (u, u) + Pf, t > 0, (4)

107

108

where Apu := P (−∆)u, P : Lp(Rd) → Lσp (Rd) is well-known as Helmholtz-Leray’s projection, and the nonlinear

term F (u, v) := −P (u · ∇)v. Equation (4) can be written as a Volterra equation of the form

u+ (ℓ ∗ Aru)(t) = u0 + (ℓ ∗ [F (u, u) + Pf ])(t), t > 0, (5)

by condition (k, ℓ) ∈ PC.

2 Main Results

We investigate the existence and uniqueness of global mild solutions for equation (5). Before we state the

main result, we introduce space where the mild solution will dwell. Let d ∈ N. For any 2 ≤ d < q < ∞,

consider the space Xq of the functions v satisfying v ∈ Cb([0,∞);Lσd (Rd)), (1 ∗ ℓ)

12−

d2q v ∈ Cb((0,∞);Lσ

q (Rd)) and

(1 ∗ ℓ) 12∇v ∈ Cb((0,∞);Lσ

d (Rd)), which is a Banach space with norm

∥v∥Xq:= maxsup

t>0∥v(t)∥Lσ

d (Rd), supt>0

[(1 ∗ l)(t)]12−

d2q ∥v(t)∥Lσ

q (Rd), supt>0

[(1 ∗ l)(t)] 12 ∥∇v(t)∥Lσ

d (Rd).

The existence of the mild solutions solution for (5) will be a consequence of the following fixed point lemma (see

[2, Lemma 1.5]).

Lemma 2.1. Let X be an abstract Banach Space and L : X×X → X a bilinear operator. Assume that there exists

η > 0 such that , given x1, x2 ∈ X, we have ∥L(x1, x2)∥ ≤ η∥x1∥∥x2∥. Then for any y ∈ X, such that 4η∥y∥ < 1,

the equation x = y + L(x, x) has a solution x in X. Moreover, this solution x is the only one such that

∥x∥ ≤1 −

√1 − 4η∥y∥2η

. (1)

Theorem 2.1. Let d ∈ N, 2 ≤ d < q < ∞, η an appropriate constant and f ∈ Cb([0,∞);L qdq+d

(Rd)) be such

that α := supt>0[(1 ∗ ℓ)(t)]1−d2q ∥f(t)∥L qd

q+d

(Rd) < ∞. For u0 ∈ Lσd (Rd) and α > 0 sufficiently small, there exists

0 < λ < 1−4αϑCη4η , where ϑ and C are positive real constants, such that if ∥u0∥Ld(Rd) ≤ min1, C−1λ, then the

problem (5) has a global mild solution u ∈ Xq that is the unique one satisfying (1). In particular,

∥u(t, ·)∥Lq(Rd) ≤1−

√1− 4η(λ+ αϑC)

2η[(1 ∗ ℓ)(t)]−

12+ d

2q and ∥∇u(t, ·)∥Ld(Rd) ≤1−

√1− 4η(λ+ αϑC)

2η[(1 ∗ ℓ)(t)]−

12 .

If, in addition, f ≡ 0, we have

[(1 ∗ ℓ)(t)]12−

d2q ∥u(t, ·)∥Lq(Rd) → 0 and [(1 ∗ ℓ)(t)] 1

2 ∥∇u(t, ·)∥Ld(Rd) → 0,

as t→ 0+. Furthermore, let u, v ∈ Xq be two solutions given by the existence part corresponding to the initial data

u0 and v0, respectively. Then,

∥u− v∥Xq≤ C√

1 − 4η(λ+ αϑC)∥u0 − v0∥Ld(Rd).

References

[1] carlone, r. and fiorenza, a. and tentarelli, l. - The action of Volterra integral operators with highly

singular kernels on Holder continuous, Lebesgue and Sobolev functions, J. Funct. Anal., 273, 1258-1294, 2017.

[2] cannone, m. - A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana,

13, 515-541, 1997.


EXISTENCIA GLOBAL E NAO GLOBAL DE SOLUCOES PARA UMA EQUACAO DO CALOR

COM COEFICIENTES DEGENERADOS

RICARDO CASTILLO1, OMAR GUZMAN-REA2 & MARIA ZEGARRA3

1Departamento de Matematica, UBB, Concepcion, Chile, [email protected],2Departamento de Matematica, Universidade de Brasılia, Brasılia-DF, Brasil, [email protected],

3Departamento de Matematica , UNMSM, Lima, Peru,[email protected]

Abstract

Neste trabalho estabelecemos condicoes para a existencia global e nao global de solucoes nao negativas

da seguinte equacao do calor ut − div(ω(x)∇u) = h(t)f(u) + l(t)g(u) em RN × (0, T ), com condicao inicial

0 ≤ u(0) = u0 ∈ C0(RN ), onde ω(x) e um peso adequado na classe Muckenhoupt A1+ 2N

que pode ter

uma linha de singularidades e (h, f, l, g) ∈ (C[0,∞))4. Quando h(t) ∼ tr (r > −1), l(t) ∼ ts (s > −1),

f(u) = (1 + u)[ln(1 + u)]p e g(u) = uq obtemos o expoente de Fujita e o segundo expoente crıtico no sentido de

Lee e Ni [5]. Nossos resultados ampliam os obtidos por Fujishima et al. [2].

1 Introducao

Consideramos a seguinte equacao do calorut − div(ω(x)∇u) = h(t)f(u) + l(t)g(u) em RN × (0, T ),

u(0) = u0 ≥ 0 em RN ,(1)

onde u0 ∈ C0(RN ), h, l ∈ C[0,∞), ω(x) e tal que

(A) ω(x) = |x1|a com a ∈ [0, 1) se N = 1, 2 e a ∈ [0, 2/N) se N ≥ 3,

(B) ω(x) = |x|b com b ∈ [0, 1), (x = (x1, ..., xN )),

e f, g ∈ C[0,∞) sao funcoes localmente Lipschitz nao negativas. Para a existencia nao global consideramos

(F1)∫∞w

dσf(σ) <∞ para todo w > 0 e f(S(t)v0) ≤ S(t)f(v0) para todo 0 ≤ v0 ∈ C0(RN ) e t > 0.

(G1)∫∞w

dσg(σ) <∞ para todo w > 0 e g(S(t)v0) ≤ S(t)g(v0) para todo 0 ≤ v0 ∈ C0(RN ) e t > 0.

Onde S(t)u0(x) :=∫RN Γ(x, y, t)u0(y)dy para t > 0, e Γ(x, y, t) e a solucao fundamental do problema homogeneo

ut − div(ω(x)∇u) = 0. Quando ω(x) = |x1|a o problema (1) esta relacionado com o laplaciano fracionario por

meio da extensao de Caffarelli-Silvestre, veja [1] e [2]. A equacao do calor (1) aparece em modelos que descrevem

processos de propagacao do calor em meios nao homogeneos, veja [3].


No presente trabalho estabelecemos condicoes para a existencia global e nao global de solucoes nao negativas de

(1). O primeiro trabalho nesta direcao foi obtido por Fujishima, Kawakami e Sire em [2], nesse trabalho os autores

obtem resultados do tipo Fujita para o problema (1), quando h = 1, l = 0 e f(u) = up (p > 1). Nosso principal

resultado e o seguinte

Teorema 2.1. Assuma a condicao (A) ou (B) e suponha que (f, g) ∈ (C[0,∞))2 sao funcoes nao negativas

localmente lipchitz contınuas tal que f(0) = g(0) = 0.

109

110

(i) Se f , g, f(s)/s, g(s)/s sao nao decrescentes num intervalo (0,m] e existe v0 = 0, 0 ≤ v0 ∈ C0(RN ),

∥v0∥∞ ≤ m tal que∫∞0h(σ) f(∥S(σ)v0∥∞)

∥S(σ)v0∥∞dσ +

∫∞0l(σ) g(∥S(σ)v0∥∞)

∥S(σ)v0∥∞dσ < 1, entao existe uma constante δ > 0

tal que para δv0 = u0 a solucao de (1) e global.

(ii) Seja 0 ≤ u0 ∈ C0(RN ), u0 = 0 e suponha que alguma das seguintes condicoes sejam satisfeitas

(a) (F1) e verdade e f e nao decrescente tal que f(s) > 0 para todo s > 0 e existe τ > 0 tal que∫∞∥S(τ)u0∥

dσf(σ) ≤

∫ τ

0h(σ)dσ.

(b) (G1) e verdade e g e nao decrescente tal que g(s) > 0 para todo s > 0 e existe τ > 0 tal que∫∞∥S(τ)u0∥

dσg(σ) ≤

∫ τ

0l(σ)dσ.

Entao a solucao de (1) com condicao inicial u0 nao e global.

Proof. Primeiro obtemos as solucoes u ∈ C((0, T ), C0(RN )) que satisfazem a formulacao integral u(x, t) =∫RN Γ(x, y, t)u0(y)dy+

∫ t

0

∫RN Γ(x, y, t− σ)h(σ)f(u(y, σ))dσ. Daı, a prova e obtida adaptando as ideias de [4] junto

com as estimativas obtidas em [2].

Agora, para ρ > 0 considere Iρ = ψ ∈ C0(RN ), ψ ≥ 0 e lim sup|x|→∞ |x|ρψ(x) <∞ e Iρ = ψ ∈ C0(RN ), ψ ≥0 e lim inf |x|→∞ |x|ρψ(x) > 0. Do teorema (2.1) e os metodos empregados em [5] e [4] obtemos os seguintes

resultados concernentes ao expoente crıtico de Fujita e ao segundo expoente crıtico no sentido de [5].

Corolario 2.1. Assuma a condicao (A) ou (B). Suponha f(t) = (1 + t)(ln(1 + t))p (p > 1), g(t) = tq (q > 1),

(h, l) ∈ (C[0,∞))2 tal que h(t) ∼ tr (r > −1) e l(t) ∼ ts (s > −1) para t suficientemente grande.

(i) Se 1 < p ≤ 1 + (2−α)(r+1)N ou 1 < q ≤ 1 + (2−α)(s+1)

N , entao o problema (1) nao tem solucoes globais nao

triviais.

(ii) Suponha 1 + (2−α)(r+1)N < p e 1 + (2−α)(s+1)

N < q.

(a) Se 0 < ρ < (2−α)(r+1)p−1 ou 0 < ρ < (2−α)(s+1)

q−1 , entao o problema (1) nao tem solucoes globais nao

triviais para qualquer condicao inicial ψ ∈ Iρ .

(b) Se (2−α)(r+1)p−1 < ρ e (2−α)(s+1)

q−1 < ρ, entao para qualquer condicao inicial ψ ∈ Iρ existe λ > 0 tal que o

problema (1) com condicao inicial λψ tem uma solucao global nao trivial.

Onde α = a quando a condicao (A) e verdade e α = b quando (B) e verdade.

References

[1] caffarelli, l. and silvestre, l. - An extension problem related to the fractional Laplacian, Commun.

Partial Differ. Equ. 32 (2007) 1245-1260. , Commun. Partial Differ. Equ. 32 (2007) 1245-1260.

[2] fujishima, y. and kawakami, t. and sire, y. - Critical exponent for the global existence of solutions to a

semilinear heat equation with degenerate coefficients., Calc. Var. Partial Differential Equations 58 (2019) 62.

[3] kamin, s. and rosenau, p. - Propagation of thermal waves in an inhomogeneous medium, Comm. Pure

Appl. Math. 34 (1981) 831-852.

[4] loayza, m. and Da Paixao, c. s. - Existence and non-existence of global solutions for a semilinear heat

equation on a general domain, Electron. J. Differential Equations, v. 2014, p. 1-9, 2014.

[5] lee, t. y. and ni w. m. - Global existence, large time behavior and life span of solutions of a semilinear

parabolic Cauchy problem,Trans. Amer. Math. Soc., v. 333, n. 1, p. 365-378, 1992.


EXISTENCIA DE ESCOAMENTOS DE FLUIDOS MAGNETICOS PERIODICOS NO TEMPO

MARIA NILDE F. BARRETO1 & JAUBER C. OLIVEIRA2

1Departamento de Matematica, UFC, CE, Brasil, [email protected],2Departamento de Matematica, UFSC, SC, Brasil, [email protected]

Abstract

Neste trabalho estabelecemos a existencia de solucoes fortes T -periodicas para um sistema de equacoes

diferenciais parciais associadas a um modelo (de Rosensweig) para o escoamento de fluıdos magneticos em

domınios limitados bidimensionais e tridimensionais sob a acao de um campo magnetico externo.

1 Introducao

Investigamos a existencia de movimentos periodicos no tempo em escoamentos de fluidos magneticos. Tais fluidos

sao encontramos em varias aplicacoes industriais como em mancais hidrostaticos e hidrodinamicos, em sistemas

de amortecimento, em atuadores e maquinas eletricas, dentre outros. Varias aplicacoes estao descritas em [1]. O

sistema de equacoes diferenciais parciais do modelo e o seguinte:

ρ(∂u∂t + (u · ∇)u

)− (η + ξ) ∆u + ∇p = µ0 (m · ∇) · h + 2ξ ∇×w em (0,∞) × Ω, (1)

∇ · u = 0 em (0,∞) × Ω, , (2)

∂m

∂t+ (u · ∇)m = w ∧m− 1

τ(m− χ0h) em (0,∞) × Ω (3)

ρ k(∂w∂t + (u · ∇)w

)− σ

′∆w − (η

′+ λ

′)∇∇ ·w = µ0 m ∧ h (4)

+2ξ (∇× u− 2w) em (0,∞) × Ω, (5)

∇ · (h + m) = F, ∇× h = 0, em (0,∞) × Ω. (6)

Estas equacoes descrevem (respectivamente) o balanco de momento linear, massa, magnetizacao, momento

angular, seguido das denominadas equacoes magneto-estaticas para o campo magnetico h. u denota a velocidade

do ferro-fluıdo, p representa a pressao dinamica, m denota a magnetizacao, e w e a velocidade de rotacao.

ρ, η, ξ, µ0, τ, χ0, k, σ′, η

′, λ

′sao constantes positivas. O campo magnetico externo e denotado por Hext e F =

−∇ · Hext. Se o termo regularizante −σ∆m e desconsiderado (desprezando-se o momento magnetico de rotacao

[5]), ate mesmo a existencia de solucoes fracas nao e conhecida. As condicoes de contorno e de periodicidade sao

as seguintes:

u = 0, w = 0, m · ν = 0, ∇×m ∧ ν = 0 sobre ∂ΩT , (7)

com a condicao de T -periodicidade para u,w,m e h. Os seguintes trabalhos anteriores abordam questoes de

existencia de solucoes para este modelo: [3],[4],[6].

111

112


Sejam

V(Ω) := φ ∈ D(Ω) : divφ = 0 , Hdiv(Ω) :=u ∈ L2(Ω) : divu ∈ L2(Ω)

com norma ∥u∥Hdiv

:=(∥u∥2 + ∥div u∥2

)1/2.

V (Ω) e o fecho de V(Ω) em H10 (Ω). Hdiv,0(Ω) e o fecho de D(Ω) em Hdiv(Ω).

Seja A o operador de Stokes, A : V (Ω) → V (Ω)∗ definido por ⟨Av,u⟩V ∗,V =∫Ω∇u : ∇v dx, ∀u,v ∈ V (Ω) com

domınio D(A) := u ∈ V (Ω) : Au ∈ H(Ω) e (u|v)D(A) := (u|v) + (Au|Av), ∀u,v ∈ D(A).

Temos a seguinte caracterizacao para o espaco Hν(Ω) = H1(Ω) ∩ N (γν), onde γν(u) = u · ν e o operador

traco, contınuo de Hdiv(Ω) em H−1/2(∂Ω). Sejam L( ) = −∆( ) com domınio D(L) = H2(Ω) ∩ Hdiv,0(Ω), e

L = −η′∆ − (η

′+ λ

′)∇div com domınio

D(L) = H2(Ω) ∩H10 (Ω).

Apresentamos neste trabalho os seguintes resultados de regularidade das solucoes fracas. O existencia de solucoes

fracas esta entre os resultados desta investigacao. Apresentamos somente resultados de regularidade destas solucoes.

Teorema 2.1. (Regularidade - caso 2d)

Seja F ∈ H1(0, T ;H1(Ω)) tal que (F |1) = 0 sobre [0, T ]. Seja Ω ⊂ R3 um subconjunto aberto limitado, simplesmente

conexo com fronteira suave (pelo menos de classe C3). Entao, as solucoes fracas T -periodicas do sistema de equacoes

do modelo de Rosensweig tem a seguinte regularidade adicional:

u ∈ L∞(0, T ;V (Ω)) ∩ L2(0, T ;D(A)), w ∈ L∞(0, T ;H10 (Ω)) ∩ L2(0, T ;D(L))

m ∈ L∞(0, T ;Hν(Ω)) ∩ L2(0, T ;D(L)), h ∈ L∞(0, T ;Hν(Ω)) ∩ L2(0, T ;H2(Ω)).

Teorema 2.2. (Regularidade - caso 3d)

Seja F ∈ H1(0, T ;H1(Ω)) tal que (F |1) = 0 sobre [0, T ]. Seja Ω ⊂ R3 um subconjunto aberto limitado, simplesmente

conexo com fronteira suave (pelo menos de classe C3). Existe uma constante positiva c tal que se ∥F∥H1(0,T ;H1) ≤ c,

entao as solucoes fracas T -periodicas do sistema de Rosensweig tem a seguinte regularidade adicional:

u ∈ L∞(0, T ;V (Ω)) ∩ L2(0, T ;D(A)), w ∈ L∞(0, T ;H10 (Ω)) ∩ L2(0, T ;D(L))

m ∈ L∞(0, T ;Hν(Ω)) ∩ L2(0, T ;D(L)), h ∈ L∞(0, T ;Hν(Ω)) ∩ L2(0, T ;H2(Ω)).

References

[1] odenbach, s. (Ed.) - Colloidal Magnetic Fluids: Basics, Development and Application of Ferrofluids, Lect.

Notes Phys. 763, Springer, Berlin Heidelberg, 2009.

[2] rosensweig, r.e. - Ferrohydrodynamics, Dover Publications, 2014.

[3] amirat, y., hamdache, k., murat, f. -Global Weak Solutions to Equations of Motion for Magnetic Fluids,

J. math. fluid mech. 10 (2008): 326-351.

[4] wang, y., tan, z. - Global existence and asymptotic analysis of weak solutions to the equations of

Ferrohydrodynamics, Nonlinear Analysis: Real World Applications, 11 (2010): 4254-4268.

[5] torrey, h.c. - Bloch Equations with Diffusion Terms, Phys. Rev., 104(3) (1956): 563-265.

[6] xie c. - Global solvability of the Rosensweig system for ferrofluids in bounded domains, Nonlinear Analysis:

Real World Applications, 48 (2019), 1-11.


ASYMPTOTIC BEHAVIOR OF THE COUPLED KLEIN-GORDON-SCHRODINGER SYSTEMS ON

COMPACT MANIFOLDS

CESAR A. BORTOT1, THALES M. SOUZA2 & JANAINA P. ZANCHETTA3

1Department of Mobility Engineering, UFSC, SC, Brazil, [email protected],2Department of Mobility Engineering, UFSC, SC, Brazil, [email protected],

3Department of Mathematics, UEM, PR, Brazil, [email protected]

Abstract

This paper is concerned with a 2-dimensional Klein-Gordon-Schrodinger system subject to two types of

locally distributed damping on a compact Riemannian manifold M without boundary. Making use of unique

continuation property, the observability inequalities, and the smoothing effect due to Aloui, we obtain exponential

stability results.

1 Introduction

In this paper, we consider the Cauchy problem of the following Klein-Gordon-Schrodinger equations through Yukawa

interaction, iψt + ∆ψ + iαBj(x, ψ) = ϕψχω in M× (0,∞), j = 1, 2

ϕtt − ∆ϕ+ a(x)ϕt = |ψ|2χω in M× (0,∞)

ψ(0) = ψ0, ϕ(0) = ϕ0, ϕt(0) = ϕ1 in M

(Pj)

where (M, g) is a bidimensional compact Riemannian manifold without boundary and g represents your metric,

Bj(x, ψ) is non-linear locally distributed damping, ω is a region on M where the dissipative effect is effective, and

χω is the characteristic function on ω. Moreover the constant α will be characterized later. We study two types of

damping, defined by B1(x, ψ) = b(x)(1 − ∆)1/2b(x)ψ and B2(x, ψ) = b(x)(|ψ|2 + 1)ψ

We assume that a(·), b(·) are non-negative functions satisfyinga, b ∈W 1,∞(M) ∩ C∞(M)

a(x) ≥ a0 > 0 in ω, and b(x) ≥ b0 > 0 in ω,

where ω is an open subset of M such that meas(ω) > 0 satisfying geometric control condition.

An interesting result of exponential decay considering Klein-Gordon-Schrodinger system with localized damping

in both equations are due the authors in [1]. Uniform decay rates were have obtained combining multipliers

method, integral inequalities of energy, and regularizing effect due to Aloui. Recently in [2], the authors generalize

the previous results considering the weaker damped structure iαb(x)(|ψ|2 + 1)ψ instead of iαb(x)(−∆)12 b(x)ψ

assumed in [1], making use of the observability inequality in both equations, the linear wave and the Schrodinger

equation, furthermore, combined with other tools have proven exponential decay. The purpose of the present article

is to extend substantially all previous results given by [1] and [2] in the geometric sense and exhibit an important

multiplier function. Here we study the problem (Pj), j = 1, 2, on a compact Riemannian manifold with arbitrary

metric.

113

114

2 Main Results

We shall use standard Sobolev spaces on Riemannian manifolds. In the present paper, we consider some crucial

assumptions about the dissipative region ω, in order to establish the geometric control condition.

Assumption 1: We assume that a, b ∈W 1,∞(M) ∩ C∞(M) are nonnegative functions such that a(x) ≥ a0 >

0 and b(x) ≥ b0 > 0 in ω. In addition, if a(x) ≥ a0 > 0 in M, then we consider χω ≡ 1 in M If b(x) ≥ b0 >

0 in M, then we consider χω ≡ 1 in M.

Definition 2.1. (Geometric Control Condition): ω geometrically controls M, i.e there exists T0 > 0, such that

every geodesic of M travelling with speed 1 and issued at t = 0, enters the set ω in a time t < T0.

Assumption 2: We assume that ω is an open subset of M such that meas(ω) > 0 and satisfying the geometric

control condition

In what follows, we consider the energy associated with problems (Pj), j = 1, 2, by

E(t) :=1

2

∫M

(|ψ(x, t)|2 + |∇ϕ(x, t)|2 + |ϕt(x, t)|2

)dM. (1)

Theorem 2.1 (Well-Posedness). Suppose Assumption 1 holds. In addition, assume that 5(2a0b0)−1 ≤ α in the

problem (P2). Then, given (ψ0, ϕ0, ϕ1) ∈ V ∩H2(M)2 ×V problems (P1) and (P2) has a unique regular solution

satisfying

ψ ∈ L∞(0,∞;V ∩H2(M)), ψ′ ∈ L∞(0,∞;L2(M)),

ϕ ∈ L∞(0,∞;V ∩H2(M)), ϕ′ ∈ L∞(0,∞;V), and ϕ

′′ ∈ L∞(0,∞;L2(M)).(2)

Considering the phase space H := V ∩ H2(M)2 × V, in the next theorem, below, we provide a local

uniform decay of the energy. Indeed, we shall consider the initial data taken in bounded sets of H, namely,

∥(ψ0, ϕ0, ϕ1)∥H ≤ L, where L is a positive constant. This is strongly necessary due to the non linear character of

system (P1) and since the energy E(t) is not naturally a non increasing function of the parameter t. Thus, the

constants, C and γ which appear below, will depend on L > 0. We shall denote d = d(c, ∥b∥∞, L), to be fixed,

where c comes from the embedding D[(1−∆)

14

]≡ H

12 (M) → L4(M). So, under the above considerations, we can

establish the main result concerning uniform stabilization from the problems (P1) and (P2).

Theorem 2.2. Suppose that the hypotheses of Theorem 2.1 holds. In addition, α >a−10 b−4

0 d2 or d is sufficiently

small. Then, there exist C, γ positive constants such that following decay rate holds E(t) ≤ Ce−γtE(0), for all t ≥ 0,

for every regular solution of problem (P1) satisfying (2), provided the initial data are taken in bounded sets of H.

Theorem 2.3. Suppose that the hypotheses of Theorem 2.1, and Assumption 2 hold. Then, there exist C, γ positive

constants such that the following decay rate holds E(t) ≤ Ce−γtE(0), for all t ≥ 0, for every regular solution of

problem (P2) satisfying (2), provided the initial data are taken in bounded sets of H.

References

[1] almeida, a. f., cavalcanti, m. m., zanchetta, j. p. - Exponential decay for the coupled Klein-Gordon-

Schrodinger equations with locally distributed damping, Communications on Pure and Applied Analysis., 17,

2039-2061, 2018.

[2] almeida, a. f., cavalcanti, m. m., zanchetta, j. p. - Exponential stability for the coupled Klein-Gordon-

Schrodinger equations with locally distributed damping, Evolution Equations and Control Theory., 8, 847-865,

2019.

[3] bortot, c. a., souza, t. m., zanchetta, j.p. - Asymptotic behavior of the cou-

pled Klein-Gordon-Schrodinger system on compact manifolds. Authorea, 2021. (Preprint,

https://doi.org/10.22541/au.162009758.85421518/v1)


GLOBAL REGULARITY FOR A 1D SUPERCRITICAL TRANSPORT EQUATION

VALTER V. C. MOITINHO1 & LUCAS C. F. FERREIRA2

1IMECC, UNICAMP, SP, Brasil, [email protected],2IMECC, UNICAMP, SP, Brasil, [email protected]

Abstract

In this manuscript we investigate a nonlocal transport 1D-model with supercritical dissipation γ ∈ (0, 1) in

which the velocity is coupled via the Hilbert transform. We show global existence of non-negative H3/2-strong

solutions in a supercritical subrange (close to 1) that depends on the initial data norm.

1 Introduction

We consider the initial value problem for the 1D transport equation with nonlocal velocity∂tθ −Hθθx + Λγθ = 0 in T× (0,∞)

θ(x, 0) = θ0(x) in T,(1)

where 0 < γ ≤ 2, T is the 1D torus, Λ = (−∆)1/2 and H denotes the Hilbert transform. This model arises as

a lower dimensional model for the well known 2D dissipative quasi-geostrophic equation and in connection with

vortex-sheet problems.

The IVP (1) has three basic cases: subcritical 1 < γ ≤ 2, critical γ = 1 and supercritical 0 < γ < 1. The global

smoothness problem in the critical and subcritical cases have already been solved (see [1, 3]).

The global regularity problem for solutions of (1) in the supercritical case remains an open problem. In the part

0 < γ < 12 of the supercritical range, Li and Rodrigo [6] proved blow-up of solutions in finite time for non-positive,

smooth, even and compactly supported initial data satisfying θ0(0) = 0 and a suitable weighted integrability

condition. In [5], still in the same range, Kiselev showed blow-up of solutions in finite time for even, positive,

bounded and smooth initial data θ0 satisfying maxx∈R θ0(x) = θ0(0) and suitable integrability conditions.

In the range 12 ≤ γ < 1, the formation of singularity in finite time or global smoothness is an open problem

(stated by [5, p. 251]), even for sign restriction on the initial data, i.e., θ0 ≥ 0 or θ0 ≤ 0. In [2], for 0 < γ < 1, Do

obtained eventual regularization of solutions for non-negative initial data.

In this work we focus on supercritical values of γ contained in the range 12 ≤ γ < 1 (close to 1) and prove

existence of global classical solutions for (1). More precisely, we show existence of H32 -strong solution for arbitrary

non-negative initial data θ0 ∈ H32 and γ1 ≤ γ < 1, where γ1 depends on H

32 -norm of θ0.

2 Main Results

Theorem 2.1. Suppose that γ ∈ [1/2, 1) and θ0 ∈ L∞(T) is non-negative. Let α ∈ (1 − γ, 1) and define

T ∗ = Cα1

1−γ ∥θ0∥γ

1−γ

L∞(T) , (1)

where C = γ−1k1kγ

1−γ

2 > 0 with k1 and k2 being independent of α, γ and θ0. Let θ be a solution of (1) in

C([0, T );H32 (T)) with existence time 0 < T <∞. If T ∗ < T , then θ ∈ C∞(T×(T ∗, T ]).

115

116

Proof. See [4].

Theorem 2.2. Let θ0 ∈ H32 (T) be an arbitrary non-negative initial data. Then, there exists γ1 = γ1(∥θ0∥

H32

) ∈(1/2, 1) such that for each γ ∈ [γ1, 1) the IVP (1) has a unique global (classical) H

32 -solution.

Proof. The Theorem 2.1 provides an explicit control on the regularization time T ∗,

T ∗ = Cα1

1−γ ∥θ0∥γ

1−γ

L∞(T) .

Afterwards, we obtain an explicit lower bound of the local existence time with H32 -initial data. More precisely,

there exists a constant C > 0 such that H32 -solution does not blow up until

T = C

(∥θ0∥

2γ(9+2γ)3(9+4γ)

L2(T) ∥θ0∥2− 4γ(6+γ)

3(9+4γ)

H32 (T)

)−1

. (2)

By comparison of the local existence time T (2) with the eventual regularization time T ∗ (1) we can choose

α = min 2(1 − γ), 1/2 such that there exists γ1 ∈ (1/2, 1) such that T ∗ < T for all γ ∈ [γ1, 1).

The equation (1) has a scaling property: if θ is a solution, then so is θλ(t, x) = λγ−1θ(λγt, λx), for any λ > 0.

Thus, the quantity ∥·∥1−2γ3

H32 (T)

∥·∥2γ3

L2(T) is scaling invariant. Now, for each γ ∈ (0, 1), define

Rγ = supR > 0 such that, for any θ0 ∈ H32 (T) with ∥θ0∥

1− 2γ3

H32 (T)

∥θ0∥2γ3

L2(T) ≤ R, the unique

H32 − solution of (1) with initial data θ0 does not blow up in finite time.

By small data results for γ ∈ (0, 1) we know that Rγ > 0, while from the global regularity results in the critical

case, we have that R1 = ∞. The Theorem 2.2 state shows that

Rγ → ∞ as γ → 1−.

References

[1] cordoba, a., Cordoba, a. and fontelos, m. - Formation of Singularities for a Transport Equation with

Nonlocal Velocity. Annals of Mathematics, 162 (3), 1377–1389, 2005.

[2] do, t. - On a 1D transport equation with nonlocal velocity and supercritical dissipation. Journal of Differential

Equations, 256 (9), 3166–3178, 2014.

[3] dong, h. - Well-posedness for a transport equation with nonlocal velocity. Journal of Functional Analysis,

255 (11), 3070–3097, 2008.

[4] ferreira, l. and moitinho, v. - Global smoothness for a 1D supercritical transport model with nonlocal

velocity. Proceedings of the American Mathematical Society, 148 (7), 2981-2995, 2020.

[5] kiselev, a. - Regularity and blow up for active scalars. Math. Model. Nat. Phenom. 5 (4), 225–255, 2010.

[6] li, d. and rodrigo, j. - Blow-up of solutions for a 1D transport equation with nonlocal velocity and

supercritical dissipation. Advances in Mathematics, 217 (6), 2563–2568, 2008.


A KATO TYPE EXPONENT FOR A CLASS OF SEMILINEAR EVOLUTION EQUATIONS WITH

TIME-DEPENDENT DAMPING

WANDERLEY NUNES DO NASCIMENTO1, MARCELO REMPEL EBERT2 & JORGE MARQUES3

1Instituto de Matematica e Estatıstica da UFRGS, [email protected],2Instituto de Matematica e Estatıstica da UFRGS,3Instituto de Matematica e Estatıstica da UFRGS

Abstract

In this presentation, we derive suitable optimal Lp−Lq decay estimates, 1 ≤ p ≤ 2 ≤ q ≤ ∞, for the solutions

to the σ-evolution equation, σ > 1, with scale-invariant time-dependent damping and power nonlinearity |u|p,

utt + (−∆)σu+µ

1 + tut = |u|p, t ≥ 0, x ∈ Rn,

where µ > 0, p > 1. The critical exponent p = pc for the global (in time) existence of small data solutions to

the Cauchy problem is related to the long time behavior of solutions, which changes accordingly with µ. Under

the assumption of small initial data in L1 ∩ L2, we find the critical exponent

pc = 1 +2σ

[n− σ + σµ]+,

for µ ∈ (0, 1). This critical exponent it is a shift of a Kato type exponent.

1 Introduction

In this presentation we study the global (in time) existence of small data solutions to the Cauchy problem for the

semilinear damped σ-evolution equations with scale-invariant time-dependent damping

utt + (−∆)σu+µ

1 + tut = |u|p, u(0, x) = 0, ut(0, x) = u1(x), t ≥ 0, x ∈ Rn. (1)

where µ ∈ (0, 1), σ > 1 and f(u) = |u|p for some p > 1.

Naturally the size of the parameter µ is relevant to describe the asymptotic behavior of solutions. When

µ ∈ (0, 1), this model is related to the semilinear free σ−evolution equations. For this reason let us introduce some

previous results to the Cauchy problem

utt + (−∆)σu = |u|p, u(0, x) = 0, ut(0, x) = u1(x). (2)

For σ = 1 this problem was considered by several authors. If 1 < p < pK(n) = n+1[n−1]+

, Kato [5] proved the

nonexistence of global generalized solutions to (2), for small initial data with compact support. On the other hand,

John [4] showed that p = 1 +√

2 is the critical exponent for the global existence of classical solutions with small

initial data in space dimension n = 3. A bit later some authors, for instance, Glassey ([2], [3]), Sideris [7], Lindblad

and Sogge [6], proved that the critical exponent is pS(n) for n ≥ 2, which is the positive root of

(n− 1)p2 − (n+ 1)p− 2 = 0.

Then, for σ > 1 and for space dimensions 1 ≤ n ≤ 2σ, in [1] it was obtained the critical exponent to (2),

pK(n) = n+σ[n−σ]+

, which is of Kato type.

The main goals in this presentation are to derive Lp − Lq estimates and energy estimates for solutions to the

linear Cauchy problem associated to (1) and to obtain the critical exponent for the global (in time) existence of

small initial data solutions to (1) for µ ∈ (0, 1). We show that the critical exponent is a shift of Kato type exponent

pK(n+ σµ).= n+σ+σµ

[n−σ+σµ]+.

117

118

2 Main Results

This two results shows that the critical exponent for the Cauchy problem (1), when µ ∈ (0, 1), is given by a shift

of Kato type exponent pK(n+ σµ).= n+σ+σµ

[n−σ+σµ]+.

Let us consider µ♯ = ∞ if µ ≤ 2 − 2nσ or µ♯ = 1

2σ

(σ − n+

√9σ2 − 10nσ + n2

)if µ > 2 − 2n

σ .

Theorem 2.1. Let σ > 1, 1 ≤ n < σ, 1 − nσ < µ < min µ♯; 1 and µ = 2 − 2n

σ . If

1 +2σ

n− σ + σµ

.= pK(n+ σµ) 0 such that for any initial data

u1 ∈ A = L2(Rn) ∩ L1(Rn), ||u1||A ≤ ϵ,

there exists a unique energy solution u ∈ C([0,∞), Hσ(Rn)) ∩ C1([0,∞), L2(Rn)) ∩ L∞([0,∞) × Rn) to (1).

Moreover, for 2 ≤ q ≤ q1 the solution satisfies the following estimates

||u(t, ·)||Lq ≲ (1 + t)−nσ (1− 1

q )+1−µ||u1||A, ||u(t, ·)||L∞ ≲ (1 + t)−minnσ+µ−1,µ2 ||u1||A,

and

∥u(t, ·)∥Hσ + ∥∂tu(t, ·)∥L2 ≲ (1 + t)−µ2 ||u1||A, ∀t ≥ 0.

For the sake of simplicity, in the next result we restrict our analysis for integer σ.

Proposition 2.1. Let σ ∈ N, 0 < µ ≤ 1 and

1 0, (4)

then there exists no global (in time) weak solution u ∈ Lploc([0,∞) × Rn) to (1).

References

[1] EBERT, M. R., LOURENCO, L. M., The critical exponent for evolution models with power non-linearity, in:

Trends in Mathematics, New Tools for Nonlinear PDEs and Applications, Birkhauser Basel, 153–177, 2019.

[2] GLASSEY, R. T., Finite-time blow-up for solutions of nonlinear wave equations. Math. Z. 177, no. 3, 323–340

(1981).

[3] GLASSEY, R. T., Existence in the large for u = F (u) in two space dimensions. Math. Z. 178, no. 2, 233–261

(1981).

[4] JHON, F., Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math. 28,

no. 1-3, 235–268, 1979.

[5] KATO, T., Blow-up of solutions of some nonlinear hyperbolic equations. Comm. Pure Appl. Math. 33, no. 4,

501–505, 1980.

[6] LINDBLAD,H., SOGGE, C., Long-time existence for small amplitude semilinear wave equations. Amer. J.

Math. 118, no. 5, 1047–1135, 1996.

[7] SIDERIS, T. C., Nonexistence of global solutions to semilinear wave equations in high dimensions. J. Differential

Equations 52, no. 3, 378–406, 1984.


EXISTENCE AND CONTINUOUS DEPENDENCE OF THE LOCAL SOLUTION OF NON

HOMOGENEOUS KDV-K-S EQUATION

YOLANDA SANTIAGO AYALA1 & SANTIAGO ROJAS ROMERO2

1Universidad Nacional Mayor de San Marcos, Fac. de Ciencias Matematicas, Lima, Peru, [email protected],2Universidad Nacional Mayor de San Marcos, Fac. de Ciencias Matematicas, Lima, Peru, [email protected]

Abstract

In this work, we prove that initial value problem associated to the nonhomogeneous KdV-Kuramoto-

Sivashinsky (KdV-K-S) equation in periodic Sobolev spaces has a local solution in [0, T ] with T > 0, and

the solution has continuous dependence with respect to the initial data and the nonhomogeneous part of the

problem. We do this in an intuitive way using Fourier theory and introducing a Co -semigroup inspired by the

work of Iorio [1] and Santiago and Rojas [3]. Also, we prove the uniqueness solution of the homogeneous and

nonhomogeneous KdV-K-S equation, using its dissipative property, inspired by the work of Iorio [1] and Santiago

and Rojas [4].

1 Introduction

First, we want to comment that from Theorem 3.1 in [2], we have that the KdV-K-S homogeneous problem is

globally well posed and, in addition to the inequality (3.2) in [2], we have the continuous dependence of the solution

of homogeneous problem.

In this work, in Theorem 2.1 we will prove the existence and uniqueness of the local solution for the non

homogeneous problem and from inequality (4) we will get the continuous dependence of the solution with respect

to the initial data and respect to the non homogeneous part.

Thus, in both homogeneous and non homogeneous cases, the estimatives are made from the explicit form of the

solution, that is, by applying the Fourier transform to the respective equation.

Another result in this work is about the dissipative property of the homogeneous problem and some estimates

of it, using differential calculus in Hsper. This is included in Theorem 2.2 which we will develop. So, using Theorem

2.2, we deduce the results of continuous dependence and uniqueness of solution for both homogeneous and non

homogeneous problems, respectively.

Finally, we give some conclusions and generalizations.

2 Main Results

We prove that the non homogeneous problem (PFc ) is locally well posed.

Theorem 2.1. Let ϕ ∈ Hsper, s ∈ R, µ > 0, F ∈ C([0, T ], Hs

per), where T > 0, and S(t)t≥0 the semigroup of

class Co of contraction in Hsper for homogeneous case (F = 0), introduced in the Theorem 3.2 from [2], then

1. The function:

uF (t) := S(t)ϕ+

∫ t

0

S(t− τ)F (τ)dτ︸︷︷︸up(t)=

, t ∈ [0, T ] (1)

belongs to C([0, T ], Hsper) ∩ C1([0, T ], Hs−4

per ) and

119

120

2. uF (t) is the unique solution of

(PFc )

∣∣∣∣∣ ut + uxxx + µ(uxxxx + uxx) = F (t) ∈ Hs−4per

u(0) = ϕ(2)

with the derivative given by

limh→0

∥∥∥∥u(t+ h) − u(t)

h+ uxxx + µ(uxxxx + uxx) − F (t)

∥∥∥∥s−4

= 0 . (3)

3. Let ψj ∈ Hsper, Fj ∈ C([0, T ], Hs

per), j = 1, 2. The map ψ −→ u is continuous in the following sense. Let u1

and u2 the corresponding solutions to initial data ψ1 and ψ2, and with non homogeneity F1 and F2 respectively.

Then

∥u1 − u2∥∞,s ≤ ∥ψ1 − ψ2∥s + T∥F1 − F2∥∞,s, (4)

∥∂tu1(t) − ∂tu2(t)∥s−4 ≤ (1 + 2µ)∥u1 − u2∥∞,s + ∥F1 − F2∥∞,s−4

≤ (1 + 2µ)∥ψ1 − ψ2∥s + [(1 + 2µ)T + 1]∥F1 − F2∥∞,s (5)

where we have used the notation:

∥h∥∞,r = supt∈[0,T ]

∥h(t)∥r , h ∈ C([0, T ], Hrper) . (6)

Now, we study the dissipative property of the homogeneous problem.

Let µ > 0, s ∈ R and the homogeneous problem

(Pc)

∣∣∣∣∣∣∣w ∈ C([0,∞), Hs

per) ∩ C1([0,∞), Hs−4per )

∂tw + ∂3xw + µ(∂4xw + ∂2xw) = 0 ∈ Hs−4per

w(0) = ϕ ∈ Hsper .

Theorem 2.2. Let w the solution of (Pc) with initial data ϕ ∈ Hsper, then we obtain the following results:

1. ∂t∥w(t)∥2s−4 = −2µ < ∂2xw(t) + ∂4xw(t), w(t) >s−4 ≤ 0.

2. ∥w(t)∥s−4 ≤ ∥ϕ∥s−4 ≤ ∥ϕ∥s, t ≥ 0.

References

[1] Iorio, R. and Iorio, V. - Fourier Analysis and Partial Differential Equations, Cambridge University, 2001.

[2] Santiago Ayala, Y. and Rojas Romero S. - Regularity and wellposedness of a problem to one parameter

and its behavior at the limit. Bulletin of the Allahabad Mathematical Society, 32(02), 207-230, 2017.

[3] Santiago Ayala, Y. and Rojas Romero S. - Existencia y regularidad de solucion de la ecuacion del calor

en espacios de Sobolev periodico. Selecciones Matematicas, 06(01), 49-65, 2019.

[4] Santiago Ayala, Y. and Rojas Romero S. - Unicidad de solucion de la ecuacion del calor en espacios de

Sobolev periodico. Selecciones Matematicas, 07(01), 172-175, 2020.


GROTHENDIECK-TYPE SUBSETS OF BANACH LATTICES

PABLO GALINDO1 & VINICIUS C. C. MIRANDA2

1Departamento de Analisis Matematico. Universidad de Valencia. Spain,2IME, USP, SP, Brazil, [email protected]

Abstract

In the setting of Banach lattices the weak (resp. positive) Grothendieck spaces have been defined. We localize

such notions by defining new classes of sets that we study and compare with some quite related different classes.

This allows us to introduce and compare the corresponding linear operators. This talk corresponds the results

in Sections 2 and 3 of the preprint [1].

1 Introduction

Recall that a Banach space X has the Grothendieck property if every weak* null sequence in E′ is weakly null.

In the class of Banach lattices, by considering disjoint and positive sequences, two further Grothendieck properties

have been considered. Following [2] (resp. [3]), a Banach lattice E has the weak Grothendieck property (resp.

positive Grothendieck property) if every disjoint weak* null sequence in E′ is weakly null (resp. every positive

weak* null sequence in E′ is weakly null). Of course the Grothendieck property implies both the positive and the

weak Grothendieck properties. The lattice c of all real convergent sequences has the positive Grothendieck property

but it fails to have the weak Grothendieck property [2, p. 10]. On the other hand, ℓ1 is a Banach lattice with the

weak Grothendieck property without the positive Grothendieck property [3, p. 6].

Recall that a subset A of a Banach space X is a Grothendieck set if T (A) is relatively weakly compact in c0 for

each bounded linear operator T : X → c0. Keeping this c0-valued operators point of view, we introduce and study a

new class of sets in Banach lattices- that we name almost Grothendieck (see Definition 2.1)- and which characterizes

the weak Grothendieck property. In an analogous way, the notion of positive Grothendieck set is defined.

2 Main Results

Every bounded linear operator T : E → c0 is uniquely determined by a weak* null sequence (x′n) ⊂ E′ such that

T (x) =(x′n(x)

)for all x ∈ E where x′n is the nth component of T. When this sequence is disjoint in the dual Banach

lattice E′, we say that T is a disjoint operator.

Definition 2.1. We say that A ⊂ E is an almost Grothendieck set if T (A) is relatively weakly compact in c0 for

every disjoint operator T : E → c0.

It is obvious that every Grothendieck set in a Banach lattice is almost Grothendieck. Obviously, each almost

Grothendieck subset of c0 is relatively weakly compact. In particular, we can localize the weak Grothendieck

property as follows:

Proposition 2.1. For a Banach lattice E, the following are equivalent:

1. E has the weak Grothendieck property.

2. Every disjoint operator T : E → c0 is weakly compact.

121

122

3. BE is an almost Grothendieck set.

By Proposition 2.1, we get that the unit ball of every L-space is an almost Grothendieck set, e.g. Bℓ1 and

BL1[0,1] are almost Grothendieck sets that are not Grothendieck sets.

The question whether the solid hull of an almost Grothendieck set is still almost Grothendieck belongs to a type

of questions usual in Banach lattice theory. In particular, we have the following results:

Theorem 2.1. Let E be a Banach lattice with the property (d) and let A ⊂ E. Then |A| = |x| : x ∈ A is almost

Grothendieck if and only if sol(A) is also almost Grothendieck.

It follows immediately from Theorem 2.1 that if A ⊂ E+ is an almost Grothendieck set in a Banach lattice with

the property (d), then sol(A) is also an almost Grothendieck set. By using this observation, we could establish

conditions so that the solid hull of an almost Grothendieck set is also an almost Grothendieck set.

Recall that a bounded operator T : X → Y is said to be Grothendieck if T (BX) is a Grothendieck set in Y . Or,

equivalently, T ′y′nω→ 0 in X ′ for every weak* null sequence (y′n) ⊂ Y ′. In a natural way, we introduce the class of

almost Grothendieck operators.

Definition 2.2. A bounded operator T : X → F is said to be almost Grothendieck if T ′y′nω→ 0 in X ′ for every

disjoint weak* null sequence (y′n) ⊂ F ′.

It is immediate that every Grothendieck operator T : X → F from a Banach space into a Banach lattice is

almost Grothendieck. The converse does not hold though. For example, the identity map Iℓ1 : ℓ1 → ℓ1 is almost

Grothendieck but not Grothendieck.

An characterization of almost Grothendieck operators concerning almost Grothendieck sets was proved as follows:

A bounded linear operator T : X → F is almost Grothendieck if and only if T (BX) is an almost Grothendieck

subset of F . As a consequence, we have that a Banach lattice F has the weak Grothendieck property if and only if

every weakly compact operator from any Banach space X into F is almost Grothendieck.

Moreover, the following result gives necessary and suficient conditions so that every almost Grothendieck set is

relatively weakly compact.

Theorem 2.2. For a Banach lattice E, every almost Grothendieck subset of E is relatively weakly compact if and

only if every almost Grothendieck operator T : X → E is weakly compact, for all Banach spaces X.

In the class of the positive linear operators in Banach lattices, there is a dominated type problem. For instance,

let S, T : E → F be positive operators such that S ≤ T . The question is, if T has some property (∗), does S also

have it? We present condition under the Banach lattice F in order to get a positive answer when T is an almost

Grothedieck operator.

By considering positive operators T : E → c0 instead of disjoint operators in Definition 2.1, we define the

positive Grothendieck sets. A study concerning this class of sets, the positive Grothendieck property and a class of

related operators was made in an analogous way.

References

[1] P. Galindo, V. C. C. Miranda, Grothendieck-type subsets of Banach lattices, arXiv:2101.06677 [math.FA]

[2] N. Machrafi, K. El Fahri, M. Moussa, B. Altin, A note on weak almost limited operators, Hacet. J. Math. Stat.

48(3) (2019), 759-770.

[3] W. Wnuk, On the dual positive Schur property in Banach lattices. Positivity 17 (2013) 759-773.


LOWER BOUNDS FOR THE CONSTANTS IN THE REAL MULTIPOLYNOMIAL

BOHNENBLUST-HILLE INEQUALITY

THIAGO VELANGA1

1Departamento de Matematica, UNIR, RO, Brasil, [email protected]

Abstract

We extend to multipolynomials the method established by [3] and [1] for finding lower bounds for the constants

in the multilinear and polynomial Bohnenblust–Hille inequalities for the case of real scalars.

1 Introduction

Given positive integers m and n1, . . . , nm, we say that a mapping P : Em → R is an (n1, . . . , nm)-homogeneous

polynomial if, for each i with 1 ≤ i ≤ m, the mapping

P (x1, . . . , xi−1, ·, xi+1, . . . , xm) : E → R

is an ni-homogeneous polynomial for all fixed xj ∈ E with j = i. Continuous multipolynomials are all those

bounded over products of the unit ball BE of E. In that case,

∥P∥ := sup |P (x1, . . . , xm)| : x1, . . . , xm ∈ BE

defines a norm on the space of all continuous (n1, . . . , nm)-homogeneous polynomials from Em into R. As for the

basics of the theory of multipolynomials between Banach spaces, we refer to [2, 4]. Similar to the polynomial case

(see [1, p. 392]), one may show that every continuous (n1, . . . , nm)-homogeneous polynomial P : c0 × · · · × c0 → Rcan be written as

P (x1, . . . , xm) =∑

cαxα11 · · ·xαm

m

for all x1, . . . , xm ∈ c0, where cα ∈ R and where the summation is taken over all matrices α ∈ Mm×∞(N0) such

that |αi| = ni, for each i with 1 ≤ i ≤ m. The multipolynomial Bohnenblust–Hille inequality [4] for real scalars

asserts that for all positive integers m and n1, . . . , nm there exists a constant CM ≥ 1 such that ∑|α1|=n1,...,|αm|=nm

|cα|2M

M+1

M+12M

≤ CM ∥P∥

for all continuous (n1, . . . , nm)-homogeneous polynomials P : c0×· · ·× c0 → R. In [3, Sec. 5], the best lower bound

for the constants in the Bohnenblust–Hille inequality for m-linear forms is given by

Cm ≥ 2m−1m (1)

for every m ≥ 2. As for the constants in the Bohnenblust–Hille inequality for m-homogeneous polynomials, the

best-obtained estimate in [1, Theorem 2.2] is given by

DR,m ≥(3

m2

)m+12m(

54

)m2

if m is even

123

124

and

DR,m ≥

(4 · 3

m−12

)m+12m

2 ·(54

)m−12

if m = 1 is odd.

In any case, we have

DR,m > (1.17)m, (2)

which holds, therefore, for every positive integer m > 1. In this work, we adapt the techniques due to [3] and [1]

aiming to yield non-trivial lower bounds for CM .

2 Main Results

In order to determine proper lower bounds for CM , let us set a couple of notations. Let f and g denote the

real-valued functions defined by means of the equations

f (ni) =

1 , if ni = 1

3ni2 , if ni is even

4 · 3ni−1

2 , if ni = 1 is odd

and

g (ni) =

1 , if ni = 1(54

)ni2 , if ni is even

2 ·(54

)ni−1

2 , if ni = 1 is odd

for each i with 1 ≤ i ≤ m.

Theorem 2.1.

CM ≥(4m−1f (n1) · · · f (nm)

)M+12M

2m−1g (n1) · · · g (nm)

for all positive integers m and n1, . . . , nm.

We conclude this study by noting that the classical multilinear and polynomial estimates can be derived from this

result. Indeed, it reduces to the best estimate (1) for the m-linear constants Cm when m > 1 and n1 = . . . = nm = 1.

An application of the theorem by assuming m = 1 and then n1 = m, on the other hand, yields the more accurate

lower bound (2) for DR,m.

References

[1] Campos, J. R., Jimenez-Rodrıguez, P., Munoz-Fernandez, G. A., Pellegrino, D., and Seoane-

Sepulveda, J. B. - On the real polynomial Bohnenblust–Hille inequality. Linear Algebra and its Applications,

465, 391-400, 2015.

[2] Chernega, I. and Zagorodnyuk, A. - Generalization of the polarization formula for nonhomogeneous

polynomials and analytic mappings on Banach spaces. Topology, 48, 197-202, 2009.

[3] Diniz, D., Munoz-Fernandez, G. A., Pellegrino, D., and Seoane-Sepulveda, J. B. - Lower bounds

for the constants in the Bohnenblust–Hille inequality: the case of real scalars. Proceedings of the American

Mathematical Society, 142, 575-580, 2014.

[4] Velanga, T. - Ideals of polynomials between Banach spaces revisited. Linear and Multilinear Algebra, 66,

2328-2348, 2018.


TIGHTNESS IN BANACH SPACES WITH TRANSFINITE BASIS

ALEJANDRA C. CACERES RIGO1

1Instituto de Matematica e Estatıstica, USP, SP, Brasil, [email protected]

Abstract

In this work we extend the definition of a tight space for Banach spaces with transfinite basis. We show some

basic properties of tight transfinite basis and we prove that a Banach space with a tight transfinite basis fails to

have minimal subspaces. Related open questions are discussed.

This research is financially supported by the FAPESP, process number 2017/18976-5.

1 Introduction

As part of the program of classification of Banach spaces up to subspaces initiated by W. T. Gowers [7], V. Ferenczi

and Ch. Rosendal [1] introduced the notion of tightness and proved several dichotomies between different notions

of tightness and minimality. Let X be a Banach space with normalized Schauder basis (xn)n. In [1] is defined a

Banach space Y to be tight in the basis (xn)n if, and only if, there is a sequence of intervals I0 < I1 < ... < Ik < ...

such that for every infinite subset A of N, we have that Y is not isomorphically embedded in the closed span

[xn : n /∈ ∪i∈AIi]. (xn)n is a tight basis for X if every Banach space Y is tight in (xn)n. A Banach space X is tight

if it has a tight basis.

In [2] it was proved that a Banach space Y is tight in the basis (xn)n if, and only if, the set

u ⊆ ω : Y is isomorphically embedded in [xn : n ∈ u] (1)

is comeager in the Cantor space 2ω, after the natural identification of P(ω) with 2ω.

H. Rosenthal defined a separable Banach space to be minimal if it can be isomorphically embedded into any of

its closed subspaces.

Tightness is hereditary by taking block subspaces meanwhile any subspace of a minimal space is always minimal.

In [1] was proved that any shrinking basic sequence of a tight space is a tight basis and that in a reflexive Banach

space every basic sequence is tight. Also, minimality and tightness are incompatible properties:

Proposition 1.1 ([1]). A tight Banach space with basis does not have minimal subspaces.

The classical spaces ℓp, c0, Schlumprecht space S [1] are minimal, meanwhile Tsirelson’s space T , the p-

convexification T p of the Tsirelson’s space [5] and Gowers-Maurey unconditional space Gu [3] are examples of

tight spaces (see [1] and [3]).

2 Main Results

We extend the notion of tightness from Banach spaces with Schauder basis to Banach spaces with transfinite basis

as follows.

Definition 2.1. Let α be an infinite ordinal. Let X be a Banach space with transfinite basis (xγ)γ<α. We say that

a Banach space Y is tight in X if, and only if,

EY := u ⊆ α : Y → [xγ : γ ∈ u]

125

126

is meager in 2α, after the natural identification of P(α) with 2α. The basis (xγ)γ<α is a tight transfinite basis for

X if, and only if, any Y Banach space is tight in X. X is tight if it admits a tight transfinite basis.

It can be proved that the property of tightness for Banach spaces with transfinite basis is hereditary by taking

transfinite block subspaces. Also, the following characterization is valid.

Proposition 2.1. Let α be an infinite ordinal and A ⊆ P(α). The following assertions are equivalent:

(i) A is comeager in 2α,

(ii) there are a sequence (In)n<ω of non-empty finite pairwise disjoint subsets of α, and subsets an ⊆ In, such

that for any u ∈ 2α, if |n : In ∩ u = an| = ℵ0, then u ∈ A.

Also, we prove that

Proposition 2.2. If (xγ)γ<α is a tight shrinking transfinite basic sequence, and (γn)n is an increasing sequence of

ordinals in α, then every basic sequence in [xγn]n is tight.

In particular, the thesis of the last proposition holds if X is a reflexive Banach space with transfinite basis

(xγ)γ<α. For spaces with transfinite basis, tightness and minimality are also incompatible properties.

Theorem 2.1. Let X be a Banach space with a tight transfinite basis, then X does not have any separable minimal

subspace.

We discuss the existence of examples of transfinite tight Banach spaces and give some open questions.

This work is part of the doctorate thesis under the supervision of the professor Valentin Ferenczi.

References

[1] ferenczi, v. and godefroy, g. - Tightness of Banach spaces and Baire category. Topics in Functional and

Harmonic Analysis,, 11, 43-55, 2011.

[2] ferenczi, v. and Rosendal, ch. - Banach spaces without minimal subspaces. Journal of Functional Analysis,

257, 149-193, 2007.

[3] ferenczi, v. and rosendal, ch. - Banach spaces without minimal subspaces - examples. Annales de l’Institut

Fourier, 62,439-475, 2011.

[4] figiel, t. and Johnson, w. - A solution to Banach’s hyperplane problem. Bulletin of the London

Mathematical Society, 29(2), 179-190, 1974.

[5] gowers, w. t. - A solution to Banach’s hyperplane problem. Bulletin of the London Mathematical Society,

26(6), 523-530, 1994.

[6] schlumprecht, t. - An arbitrarily distortable Banach space. Israel Journal of Mathematics, 156(3), 797-833,

2002.

[7] gowers, w. t. - An infinite Ramsey theorem and some Banach-space dichotomies.Annals of Mathematics,

76, 81-95, 1991.


RESULTS ON THE FRECHET SPACE HL(BE)

LUIZA A. MORAES1 & ALEX F. PEREIRA2

1Instituto de Matematica, UFRJ, RJ, Brasil, [email protected],2Instituto de Matematica e Estatıstica, UFF, RJ, Brasil, [email protected]

Abstract

For a complex Banach algebra E, in this work we study topological properties of the space HL(BE) of the

mappings f : BE −→ E that are analytic in the sense of Lorch endowed with the topology τb where BE denote

the open unit ball in E. Also, we show that HL(BE) is homeomorphic to some sequence space in E.

1 Introduction

For a commutative Banach algebra E let BE be the open unit ball of E. We denote by Γ(BE) the set of the

sequences (an)n in E that satisfy lim sup ∥an∥1/n ≤ 1. We consider Γ(BE) endowed with the topology τ generated

by the family of the seminorms ∥a∥r = sup ∥an∥rn for all a = (an)n ∈ Γ(BE) and for all 0 < r < 1. It is easy to

see that (Γ(BE), τ) is a locally convex space with the usual operations.

We say that f : BE −→ E is Lorch analytic in BE if and only if there exist unique sequence (an)n ∈ Γ(BE) such

that f(w) =∑∞

n=0 anwn for all w ∈ BE . We denote by HL(BE) the space of Lorch analytic mappings from BE

into E. For more information about Lorch analytic mappings we refer to [2]. It is clear that HL(BE) ⊂ Hb(BE , E)

where Hb(BE , E) denotes the space of holomorphic mappings from BE into E which are bounded on the bounded

subsets of BE . For background on holomorphic mappings between Banach spaces see [1, 7]. Note that we can

consider in HL(BE) the topology τb of the uniform convergence on the bounded subsets of BE .

2 Main Results

For n ∈ N0 we denote by PL(nE) the space of the n-homogeneous polynomials from E into E which are Lorch

analytic in BE with the usual topology. The proofs of the propositions below can be found in [3].

Proposition 2.1. The following statements are true:

(a) PL(nE)n∈N0 is an 1-Schauder decomposition of (HL(BE), τb).

(b) PL(nE)n∈N0 is an S-absolute decomposition of (HL(BE), τb).

(c) PL(nE)n∈N0 is shrinking.

(d) PL(nE)n∈N0 is boundedly complete.

Proposition 2.2. E has the Schur property if and only if (HL(BE), τb) has the Schur property.

Proposition 2.3. E is separable if and only if (HL(BE), τb) is separable.

Proposition 2.4. E is reflexive if and only if (HL(BE), τb) is reflexive.

Proposition 2.5. (HL(BE), τb) is a Frechet space.

It is clear by the definition of Γ(BE) that we have a natural linear bijection between Γ(BE) and HL(BE).

Theorem 2.1. (Γ(BE), τ) e (HL(BE), τb) are isomorphics. In particular, (Γ(BE), τ) is a Frechet space.

127

128

References

[1] dineen, s. - Complex Analysis on Infinite Dimensional Spaces, Springer Monogr. Math., Springer Verlag,

London, Berlin, Heidelberg, 1999.

[2] lorch, e.r. - The theory of analytic functions in normed abelian vector rings, Trans. Amer. Math. Soc., 54

(1943), pp. 414-425.

[3] mauro, g.v.s., moraes, l.a and pereira, a.f. - Topological and algebraic properties of spaces of Lorch

analytic mappings, Mathematische Nachrichten, 289 (2016), pp. 845-853.

[4] moraes, l. a and pereira, a. f. - The spectra of algebras of Lorch analytic mappings, Topology, 48 (2009),

pp. 91-99.

[5] moraes, l.a and pereira, a.f. - The Hadamard product in the space of Lorch analytic mappings, Publ.

RIMS Kyoto Univ., 47 (2013), pp. 111-122.

[6] moraes, l.a and pereira, a.f. - Duality in spaces of Lorch analytic mappings, Quarterly Journal of

Mathematics, 67 (2016), pp. 431-438.

[7] mujica, j. - Complex analysis in Banach spaces, North-Holland Math. Studies 120, Amsterdam, 1986.


SOBRE UMA REFORMULACAO DA HIPOTESE DE RIEMANN NO ESPACO DE HARDY DO

CIRCULO UNITARIO

CHARLES F. DOS SANTOS1

1IMECC, UNICAMP, SP, Brasil, [email protected]

Abstract

Este trabalho busca avancar o esforco de pesquisa iniciado pela versao recente do criterio de Nyman-Beurling-

Baez-Duarte para a hipotese de Riemann (RH, da sigla em inglos) no espaco de Hardy-Hilbert do disco unitario,

H2. Questoes de densidade e ortogonalidade diretamente atreladas a este criterio sao abordadas, o que leva

a versoes fracas de RH. Entre as ferramentas utilizadas, se destacam varios espacos de Hilbert de funcoes

holomorfas no disco unitario. Trabalho em colaboracao com J. C. Manzur.

1 Introducao

A hipotese de Riemann e a afirmacao de que a funcao definida por ζ(s) =∑∞

n=1 n−s para Re s > 1 e estendida

analiticamente a C\1 nao possui zeros com parte real maior do que 1/2. Nyman [4] obteve uma reformulacao

de RH em termos de densidade e aproximacao em L1(0, 1), resultado generalizado para Lp(0, 1) por Beurling [2]

e refinado por Baez-Duarte [1]. Detalhes podem ser encontrados no artigo expositorio [2]. Em [5], e encontrada a

seguinte versao unitariamente equivalente do criterio de Baez-Duarte. O contexto e uma classe de funcoes holomorfas

no disco unitario D, a saber o espaco de Hardy-Hilbert

H2 =

f : D → C : f(z) =

∞∑n=0

f(n)zn,

∞∑n=0

|f(n)|2 <∞

,

que e um espaco de Hilbert com a norma ∥f∥ =(∑∞

n=0 |f(n)|2)1/2

e contem o espaco H∞ das funoes holomorfas

limitadas em D. Toda f ∈ H2 possui limites radiais em quase todo ponto do cırculo unitario T, o que identifica H2

com um subespaco fechado de L2(T).

Teorema 1.1. Seja N o espaco vetorial gerado por hk : k ≥ 2, onde

hk(z) =1

1 − zlog

(1 + z + · · · + zk−1

k

), z ∈ D, k ≥ 2 .

Entao a hipotese de Riemann e verdadeira se e somente se N e denso em H2, o que ocorre se e somente se a

constante 1 esta no fecho de N .

Este trabalho fortalece resultados de [5] dando respostas parciais aos problemas de encontrar (i) topologias em

H2 com respeito as quais N e denso; (ii) subespacos vetoriais V ⊂ H2 tais que N⊥ ∩ V = 0. Tais respostas

parciais podem ser interpretadas como versoes fracas da hipotese de Riemann, ou seja, afirmacoes que sao implicadas

por RH mas demonstradas verdadeiras incondicionalmente.


Uma funcao f ∈ H2 e exterior se znf : n ≥ 0 gera um subespaco denso em H2. Funcoes exteriores nao se anulam

no disco. A classe de Smirnov e

N+ = g/h : g, h ∈ H∞, h e exterior .

129

130

Em [8] e estudada a topologia induzida em N+ pela metrica

d(f, g) =1

2π

∫ 2π

0

log(1 + |f(eiθ) − g(eiθ)|

)dθ ,

que e mais fraca que a topologia da norma e mais forte que a da convergencia uniforme em compactos.

Teorema 2.1. Com respeito a metrica d, N e denso em N+, e portanto em H2.

Dado ξ ∈ T, o espaco local de Dirichlet em ξ e

Dξ =

f : D → C : f e holomorfa,

∫D|f ′(z)|2 1 − |z|2

|ξ − z|2dA(z) <∞

,

onde dA e a medida de area. Temos que Dξ coincide com (ξ − z)f + c : f ∈ H2, c ∈ C (ver [4]) e contem

todas as funcoes holomorfas em vizinhancas do fecho de D. Usando resultados de [7] relacionando operadores de

multiplicacao ilimitados em H2 e espacos de de Branges-Rovnyak, e possıvel provar o seguinte.

Teorema 2.2. Para todo ξ ∈ T, N⊥ ∩ Dξ = 0.

References

[1] baez-duarte, l. - A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis. Atti

della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei.

Matematica e Applicazioni, 14, 5-11, 2003.

[2] bagchi, b. - On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis.

Proceedings of the Indian Academy of Sciences - Mathematiccal Sciences, 116(2), 137-146, 2006.

[3] beurling, a. - A closure problem related to the Riemann zeta-function. Proceedings of the National Academy

of Sciencesof the United States of America, 41(5), 312-314, 1955.

[4] costara, c. and ransford, t. - Which de Branges-Rovnyak spaces are Dirichlet spaces (and vice versa)?.

Journal of Functional Analysis, 265(12), 3204-3218, 2010.

[5] noor, s. w. - A Hardy space analysis of the Baez-Duarte criterion for the RH. Advances in Mathematics, 350,

242-255, 2019.

[6] nyman, b. - On some groups and semigroups of translations, Thesis, Upsalla, 1950.

[7] sarason, d. - Unbounded Toeplitz operators. Integral Equations and Operator Theory, 61(2), 281-298, 2008.

[8] yanagihara, n. - Multipliers and Linear Functionals for the Class N+. Transactions of the American

Mathematical Society , 180, 449-461, 1973.


TEOREMAS DO TIPO BANACH-STONE PARA ALGEBRAS DE GERMES HOLOMORFOS EM

ESPACOS DE BANACH

DANIELA M. VIEIRA1

1Instituto de Matematica e Estatıstica, USP, SP, Brasil, [email protected]

Abstract

Neste trabalho, estudamos resultados do tipo Banach-Stone para algebras de germes holomorfos em espacos

de Banach. Mostramos que se K e L sao subconjuntos compactos, equilibrados e determinantes em espacos

de Banach separaveis com propriedade de aproximacao, entao as algebras de germes holomorfos H(K) e

H(L) sao topologicamente isomorfas se, e somente se, as envoltorias polinomialmente convexas KP e LP sao

biholomorficamente equivalentes.

1 Introducao

Se K e L sao espacos topologicos Hausdorff compactos, o Teorema de Banach-Stone classico afirma que os espacos

C(K) e C(L) sao isometricos se, e somente se, K e L sao homeomorfos. Mais especificamente, se T : C(K) → C(L)

e uma isometria, entao existem um homeomorfismo φ : L → K e uma funcao contınua α : L → K com |α(y)| = 1,

para todo y ∈ L tais que: T (f)(y) = a(y) · (f φ)(y) = a(y) · f(φ(y)), para todo y ∈ L e para toda f ∈ C(K).

A versao desse teorema para isomorfismos algebricos foi provada por Gelfand & Kolmogoroff em 1939 e afirma

que C(K) e C(L) sao isomorfas como algebras se, e somente se, K e L sao homeomorfos. Alem disso, todo isomorfismo

algebrico T : C(K) → C(L) e da forma T (f) = f φ, onde φ : L→ K e um homeomorfismo.

Neste trabalho, investigamos resultados deste tipo em algebras de germes holomorfos em espacos de Banach. A

seguir, daremos algumas definicoes.

Sejam E um espaco de Banach complexo, U um subconjunto aberto de E. Denotamos por: H(U) o espaco

vetorial de todas as funcoes holomorfas f : U → C e τω a topologia de Nachbin (portada por compactos) em H(U).

Desta forma, (H(U), τω) e uma algebra localmente m-convexa.

Se K e um subconjunto compacto de E, denotamos por h(K) = ∪U⊃KH(U), onde U percorre todos os abertos

de E que contem K. Diremos que duas funcoes f1, f2 ∈ h(K) so equivalentes (f1 ∼ f2) se elas coincidirem em

alguma vizinhanca aberta de K. Denotamos por H(K) ao conjunto de todas as classes de equivalencias de funcoes

que sao holomorfas em alguma vizinhanca de K. Cada elemento de H(K) e chamado de germe holomorfo em

K. As aplicacoes canonicas IU : H(U) → H(K), com U ⊃ K, induzem uma estrutura de espaco vetorial em

H(K). O espaco vetorial H(K) e entao munido da topologia indutiva com respeito as aplicacoes lineares canonicas

IU : (H(U), τω) → H(K), com U ⊃ K. Desta forma, dizemos que H(K) e o limite indutivo dos espacos (H(U), τω),

com U ⊃ K. Tem-se que (H(K), τω) e uma algebra localmente m-convexa.

A envoltoria polinomialmente convexa de K e definida por

KP(E) = x ∈ E : |P (x)| ≤ supK

|P |, para todo P ∈ P(E).

Um compacto K e polinomialmente convexo se KP(E) = K. Dizemos que um subconjunto compacto K de

um espaco de Banach E e determinante se f ∈ H(U) e tal que f |K = 0 entao existe uma vizinhanca V ⊃ K,

K ⊂ V ⊂ U tal que f |V = 0. Um espaco de Banach E possui um compacto determinante se, e somente se, E e

separavel.

131

132

Sejam E e F espacos de Banach, e sejam K ⊂ E e L ⊂ F subconjuntos compactos. Dizemos que K e L sao

biholomorficamente equivalentes se existem abertos U e V com K ⊂ U ⊂ E e L ⊂ V ⊂ F , e uma aplicacao

φ : V → U biholomorfa com φ(L) = K.


Nosso principal resultado e o seguinte teorema.

Teorema 2.1. Sejam E e F espacos de Banach, ambos separaveis e com a propriedade de aproximacao, e sejam

K ⊂ E e L ⊂ F subconjuntos compactos, equilibrados e determinantes. Entao as seguintes afirmacoes sao

equivalentes:

(1) H(K) e H(L) sao topologicamente isomorfas como algebras.

(2) KP(E) e LP(F ) sao biholomorficamente equivalentes.

Esse teorema esta relacionado com um resultado semelhante de [3], para compactos equilibrados em espacos de

Banach do tipo Tsirelson. O Teorema 2.1 melhora o resultado de [3] para uma classe muito mais ampla de espacos

de Banach, porem a classe de compactos e reduzida aos compactos equilibrados e determinantes.

A demonstracao do Teorema 2.1 esta baseada em resultados de [5], tecnicas de [1, 2, 3], alem da seguinte

proposicao:

Proposition 2.1 (J. Mujica, D.M.V., 2017). Sejam E e F espacos de Banach, ambos com a propriedade de

aproximacao, e sejam K ⊂ E e L ⊂ F subconjuntos compactos e polinomialmente convexos. Se H(K) e H(L) sao

topologicamente isomorfas como algebras, entao K e L sao homeomorfos.

Como consequencia do Teorema 2.1, temos o seguinte corolario:

Corollary 2.1. Sejam E e F espacos de Banach, ambos separaveis e com a propriedade de aproximacao, e sejam

K ⊂ E e L ⊂ F subconjuntos compactos, equilibrados, determinantes e polinomialmente convexos. Entao H(K) e

H(L) sao topologicamente isomorfas como algebras se, e somente se, K e L sao biholomorficamente equivalentes.

References

[1] carando, d. and muro, s. - Envelopes of holomorphy and extension of functions of bounded type, Adv. Math.

229 (2012) 2098-2121.

[2] dineen, s. - Complex Analysis in Infinite Dimensional Spaces, Springer-Verlag, London, 1999.

[3] garcıa, d., maestre, m. and vieira, d. m. - On the Banach-Stone theorem for algebras of holomorphic

germs., Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Mat. RACSAM 111 (2017), 223-230.

[4] mujica, j. - Complex Analysis in Banach Spaces, North-Holland Math. Stud. 120, Amsterdam, 1986.

[5] waelbroeck, l - Weak analytic functions and the closed graph theorem, Lecture Notes in Math., Vol. 364,

Springer, Berlin, 1974, pp. 97-100.


OPERADORES MULTILINEARES SOMANTES E CLASSES DE SEQUENCIAS

GERALDO BOTELHO1 & DAVIDSON FREITAS2

1Faculdade de Matematica, UFU-MG,, Brasil, [email protected],2IMECC, UNICAMP-SP, Brasil, [email protected]

Abstract

Construımos neste trabalho um aparato geral que recupera como casos particulares todas as classes de

operadores absolutamente somantes definidos ou caracterizados por transformacao de sequencias vetoriais ja

estudadas na literatura.

1 Introducao

O estudo de classes de operadores multilineares que generalizam os ideais dos operadores lineares (p; q)-somantes

tem se desenvolvido nas ultimas decadas seguindo varias linhas, entre elas: (i) estudar as classes de operadores

que podem ser definidas ou caracterizadas por meio de transformacao de sequencias vetoriais; (ii) considerar as

varias formas de se percorrer o conjunto de ındices na definicao dos operadores, como por exemplo, somar apenas

na diagonal de Nd (operadores absolutamente somantes) ou somar em todos os ındices de Nd (operadores multiplo

somantes); (iii) considerar conjuntos de ındices intermediarios a diagonal e a Nd e, por fim, (iv) considerar somas

iteradas sobre conjuntos de ındices de Nd (caso anisotropico).

O objetivo deste trabalho e introduzir um conceito que unifica todas essas linhas que foram estudadas

separadamente. Cada um dos casos ate agora ja estudados sera um caso particular do conceito aqui introduzido.

Usaremos a nocao de classe de sequencias vetoriais, introduzido em [4]. Assim, dados uma classe de sequencias

X e um espaco de Banach E, X(E) sera um espaco de sequencias a valores em E, de acordo com [4]. As letras

X1, . . . , Xd, Y1, . . . , Yk denotarao classe de sequencias vetoriais, onde d e k sao numeros naturais fixos com 1 ≤ k ≤ d.


Usaremos as letras E1, . . . , Ed, F para denotar espacos de Banach, denotaremos por I = I1, . . . , Ik uma particao

do conjunto 1, . . . , d, ou seja, uma classe de subconjuntos de 1, . . . , d dois a dois disjuntos cuja uniao A© igual

a 1, . . . , d, e por x ∗ ej entenderemos a d-upla (0, . . . , 0, x, 0, . . . , 0) com x na coordenada j e 0 nas demais, seja

quando x pertencer a um espaco de Banach seja quando x for um numero natural.

Definition 2.1. Fixadas uma particao I = I1, . . . , Ik e d sequencias de numeros naturais (jrn)∞n=1, r = 1, . . . , d,

tais que a correspondencia (n1, . . . , nk) ∈ Nk 7−→∑k

s=1

∑r∈Is

jrns∗ er e injetiva, o bloco de Nd associado a particao

I e as sequencias (jrn)∞n=1, r = 1, . . . , d, e definido por BI =∑k

s=1

∑r∈Is

jrns∗ er ∈ Nd : n1, . . . , nk ∈ N

.

A notacao Y1(Y2(F )) refere-se a todas as sequencias em Y2(F ) que pertencem a classe Y1. Assim, iteradamente,

construımos o espaco Y1(· · ·Yk(F ) · · · ).

Definition 2.2. Um operador d-linear A : E1 × · · · × Ed −→ F e dito parcialmente (BI ;X1, . . . , Xd;Y1, . . . , Yk)-

somante se

(· · ·(A(∑k

s=1

∑r∈Is

xrjrns∗ er

))∞nk=1

. . .

)∞

n1=1

∈ Y1(· · ·Yk(F ) · · · ) para quaisquer sequencias (xrj)∞j=1 ∈

Xr(Er), r = 1, . . . , d. Por LBIX1,...,Xd;Y1,...,Yk

(E1, . . . , Ed;F ) denotamos a classe formada por todos esses operadores.

133

134

Proposition 2.1. Um operador d-linear A : E1 × · · · × Ed −→ F e parcialmente (BI ;X1, . . . , Xd;Y1, . . . , Yk)-

somante se, e somente se, o operador induzido ABI : X1(E1) × · · · ×Xd(Ed) −→ Y1(· · ·Yk(F ) · · · ), definido por

ABI

((x1j )∞j=1, . . . , (x

dj )∞j=1

)=

· · ·

(A

(k∑

s=1

∑r∈Is

xrjrns∗ er

))∞

nk=1

· · ·

∞

n1=1

,

esta bem definido, e d-linear e contınuo. Neste caso define-se a norma parcialmente (BI ;X1, . . . , Xd;Y1, . . . , Yk)-

somante de A por

∥A∥BI ;X1,...,Xd;Y1,...,Yk:= ∥ABI∥.

Definition 2.3. Dizemos que uma d+ k-upla de classes de sequencias (X1, . . . , Xd, Y1, . . . , Yk) e BI-compatıvel se

o operador IKd : Kd −→ K definido por IKd

(λ1, . . . , λd

)=∏d

r=1 λr e parcialmente (BI ;X1, . . . , Xd;Y1, . . . , Yk)-

somante com ∥IKd∥BI ;X1,...,Xd;Y1,...,Yk= 1.

Teorema 2.1. Sejam BI ⊆ Nd um bloco associado a particao I e as sequencias (jrn)∞n=1, r =

1, . . . , d, e X1, . . . , Xd, Y1, . . . , Yk classes de sequencias linearmente estaveis que sao BI-compatıveis. Entao

(LBIX1,...,Xd;Y1,...,Yk

, ∥ · ∥BI ;X1,...,Xd;Y1,...,Yk) e um ideal de Banach de operadores multilineares.

Neste ambiente recuperamos todas as classes que foram recuperadas em [5] e tambem outras classes que la nao

eram recuperadas, em particular o caso geral dos operadores estudados em [4]. Vejamos alguns exemplos concretos.

Exemplo 2.1. A diagonal D(Nd) := (j, d. . ., j) : j ∈ N em Nd e um bloco associado a particao trivial

I = 1, . . . , d e as sequencias (jrn)∞n=1 = (n)∞n=1, r = 1, . . . , d. Os operadores parcialmente (D(Nd), ℓwp1(·), . . . ,

ℓwpd(·); ℓq(·))-somantes sao exatamente os operadores absolutamente (q; p1, . . . , pd)-somantes de [1].

Exemplo 2.2. Seja BI o bloco associado a particao I = I1, . . . , Ik e as sequencias de numeros naturais (jrn)∞n=1 =

(n)∞n=1. Os operadores que sao parcialmente (BI ; ℓwp1(·), . . . , ℓwpd

(·); ℓq1(·), . . . , ℓqk(·))-somantes sao exatamente

os operadores que sao I-parcialmente multipo (q;p)-somante com (q;p) := (p1, . . . , pd, q1, . . . , qk) ∈ [1,∞)d+k

estudados em [2].

Exemplo 2.3. Considere Nd como o bloco associado a particao I = I1, . . . , Id, com Ir = r, e as sequencias

(jrn)∞n=1 = (n)∞n=1, r = 1, . . . , d. Entao a classe dos operadores que sao parcialmente (Nd; ℓwp1(·), . . . , ℓwpd

(·); ℓq(·), d. . .

, ℓq(·))-somantes coincide com a classe de operadores que sao multiplo (q; p1, . . . , pd)-somantes (ver [2, 3]).

References

[1] Alencar, R. and Matos, M. Some Classes of Multilinear Mappings Between Banach Spaces. Publicaciones del

Departamento de Analisis Matematico de la Universidad Complutense de Madrid, 12, 1989.

[2] Araujo, G. Some classical inequalities, summability of multilinear operators andstrange functions. PhD Thesis,

UFPB, Joao Pessoa, 2016.

[3] Bayart, F., Pellegrino, D. and Rueda, P. On coincidence results for summing multilinear operators:

interpolation,ℓ1-spaces and cotype. Collectanea Mathematica 71, 2 (2020), 301-318.

[4] Botelho, G. and Campos, J. R. On the transformation of vector-valued sequences by linear and multilinear

operators. Monatshefte fur Mathematik 183, 3 (2017), 415-435.

[5] Botelho, G. and Freitas, D., Summing multilinear operators by blocks: The isotropic and anisotropic cases.

Journal of Mathematical Analysis and Applications 490 (2020), no. 1, 124203, 21 pp.


PROPRIEDADE DE SCHUR POLINOMIAL POSITIVA

GERALDO BOTELHO1 & JOSE LUCAS PEREIRA LUIZ2

1Faculdade de Matematica, UFU-MG,, Brasil, [email protected],2IFNMG - Campus Arinos, MG, Brasil, [email protected]

Abstract

Definimos e demonstramos alguns resultados sobre a propriedade de Schur polinomial positiva. Essa

propriedade surge como um analogo, no ambiente de reticulados de Banach, a propriedade de Schur polinomial

(Λ-espaco) em espacos de Banach.

1 Introducao

O estudo da propriedade de Schur polinomial teve inıcio em 1989 com o celebre artigo de Carne, Cole e Gamelin [2],

trabalho no qual foi apresentado o conceito de Λ-espaco e foram apresentados os primeiros resultados sobre esse tipo

de espaco de Banach. Posteriormente esse conceito foi abordado por outros matematicos (veja [3, 4, 5]) e os termos

espaco polinomialmente de Schur e espaco com a propriedade de Schur polinomial passaram a ser empregados como

sinonimos de Λ-espaco.

Definicao 1.1. Um espaco de Banach X tem a propriedade de Schur polinomial se toda sequencia polinomialmente

nula em X e nula em norma.

e natural ponderar sobre uma versao analoga a propriedade de Schur polinomial em reticulados de Banach que

leve em consideracao as peculiaridades advindas da estrutura de ordem. Isso nos motiva a introduzir a seguinte

definicao:

Definicao 1.2. Um reticulado de Banach E tem a propriedade de Schur polinomial positiva se toda sequencia

positiva (xj)∞j=1 em E tal que P (xj) −→ 0 para todo polinomio homogeneo regular P em E e nula em

norma. Um reticulado de Banach com a propriedade de Schur polinomial positiva sera chamado de positivamente

polinomialmente de Schur (PPS).


A seguir listamos alguns exemplos e resultados sobre reticulados de Banach positivamente polinomialmente de

Schur, os quais podem ser encontrados em [1].

Exemplo 2.1. (a) Todo reticulado de Banach E com a propriedade de Schur positiva e PPS.

(b) L1[0, 1] e um reticulado de Banach PPS que nao tem a propriedade de Schur polinomial.

(c) O reticulado de Banach

(⊕n∈N

ℓn∞

)1

e PPS, pois tem a propriedade de Schur positiva, e nao e um AL-espaco.

Proposicao 2.1. Seja E um reticulado de Banach com a propriedade de Dunford-Pettis e sem a propriedade de

Schur positiva. Entao E nao e PPS. Em particular, AM-espacos nao sao PPS.

Exemplo 2.2. C(K)-espacos, em particular c0, nao sao PPS.

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136

A propriedade de Schur polinomial positiva e herdada por subreticulados fechados e preservada por isomorfismo

de reticulados, conforme a proposicao a seguir.

Proposicao 2.2. (a) Se F e um reticulado de Banach positivamente isomorfo a um subespaco de um reticulado

de Banach PPS, entao F e PPS.

(b) Se dois reticulados de Banach sao isomorfos como reticulados e um deles e PPS, entao o outro tambem sera

PPS.

(c) Subreticulados fechados de reticulados de Banach PPS sao PPS.

A seguir vemos que reticulados de Banach PPS gozam de boas propriedades.

Proposicao 2.3. Todo reticulado de Banach PPS e um KB-espaco, consequentemente, tem norma ordem-contınua,

e fracamente sequencialmente completo e e Dedekind completo.

O proximo exemplo nos mostra que nao vale a recıproca da proposicao anterior.

Exemplo 2.3. O espaco de Tsirelson original T ∗ e um KB-espaco que nao e PPS.

Notamos que, na realidade, os espacos Lp(µ) gozam de uma propriedade mais forte do que ser PPS.

Definicao 2.1. Dado n ∈ N, um reticulado de Banach E tem a propriedade de Schur n-polinomial positiva se toda

sequencia (xj)∞j=1 em E fracamente nula e positiva tal que P (xj) −→ 0 para todo polinomio n-homogeneo regular

P em E e nula em norma. Nesse caso diremos que E e um reticulado de Banach n-PPS.

Teorema 2.1. Sejam 1 ≤ p < ∞ e µ uma medida qualquer. O reticulado de Banach Lp(µ) e n-PPS para todo

n ≥ p.

Esse teorema nos apresenta exemplos de reticulados de Banach PPS que nao possuem a propriedade de Schur

positiva.

Corolario 2.1. Todos os reticulados de Hilbert e ℓp, 1 ≤ p <∞, sao PPS.

Para finalizar, observamos que a propriedade de Schur polinomial positiva possui uma interessante relacao com

as propriedades de Schur positiva e de Dunford-Pettis fraca, conforme teorema a seguir.

Teorema 2.2. Um reticulado de Banach tem a propriedade de Schur positiva se, e somente se, ele tem as

propriedades de Dunford-Pettis fraca e de Schur polinomial positiva.

References

[1] Botelho, G. and Luiz, J. L. P. - The positive polynomial Schur property in Banach lattices, Proc. Amer.

Math. Soc. 149 (2021), 2147–2160.

[2] Carne, T. K., Cole, B. and Gamelin, T. W. - A uniform algebra of analytic functions on a Banach space,

Trans. Amer. Math. Soc. 314 (1989), 639–659.

[3] Farmer, J. and Johnson, W. B. - Polynomial Schur and polynomial Dunford-Pettis properties, Contemp.

Math. 144 (1993), 95–105.

[4] Garrido, M. I., Jaramillo, J. A. and Llavona, J. G. - Polynomial topologies on Banach spaces, Topology

Appl. 153 (2005), 854–867.

[5] Jaramillo, J. A. and Prieto, A. - Weak-polynomial convergence on a Banach space, Proc. Amer. Math.

Soc. 118 (1993), 463–468.


A SEMIGROUP RELATED TO THE RIEMANN HYPOTHESIS

JUAN C. MANZUR1


Abstract

According to S. Waleed Noor, the cyclic vector of a semigroup of weighted composition operators are

intimately related to the Riemann hypothesis. In this work we focus on the analysis of this semigroup. In

particular, a new reformulation for the Riemann hypothesis says that the study of invariant subspaces of any

element of this semigroup are also related to this conjecture. We also provide a generalization for the BA¡ez-

Duarte criterion in H2 trough cyclic vectors.

1 Introduction

The Riemann hypothesis is a famous open problem, which says that all the non-trivial zeros of the ζ-function lie

on the vertical line with real part 1/2. This conjecture is considered to be the most important unsolved problem in

mathematics.

In 1950, [2, 4], Nyman and Beurling gave a reformulation for this problem: they proved that the Riemann

hypothesis holds if and only if the constant 1 belongs to the closure linear span of fλ : 0 < λ ≤ 1 in L2(0, 1),

where fλ(x) = λ/x−λ1/x; here x denotes the fractional part of a real number x. In 2003, [1], BA¡ez-Duarte

showed a stronger version: the family fλ : 0 < λ ≤ 1 was replaced by the countable family f1/k : k ≥ 1.

Recently, S. Waleed Noor, [7], gave the H2 version of the BA¡ez-Duarte reformulation:

Theorem 1.1. For each k ≥ 2, define

hk(z) =1

1 − zlog

(1 + z + · · · + zk−1

k

).

Then the Riemann hypothesis holds if and only if the constant 1 belongs to the closure linear span of hk : k ≥ 2in H2.

S. Waleed Noor also construted a semigroup Wn : n ≥ 1 on H2, where Wnf(z) = (1 + z + · · · + zn−1)f(zn),

of weighted composition operators having a closed relation with the Riemann hypothesis. He showed that the

constant 1 appearing in Theorem 1.1 may be replaced by any cyclic vector of Wn : n ≥ 1. So the generalization

of Theorem 1.1 was stated as follows.

Theorem 1.2. The following statements are equivalents:

1) Riemann hypothesis,

2) the closed linear span of hk : k ≥ 2 contains a cyclic vector of Wn : n ≥ 1,

3) the closed linear span of hk : k ≥ 2 is dense in H2.

This semigroup Wn : n ≥ 1 is also related to another important problem: To characterize all the 2-periodic

functions ϕ on (0,∞) having the property that the span of its dilates ϕ(nx) : n ≥ 1 is dense in L2(0, 1). This

open problem is known as the Periodic Dilation Completeness problem (PDCP).

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138

N. Nikolski, [3], proved that solving this problem is equivalent to characterizing the cyclic vectors of a semigroup

Tn : n ≥ 1 on H20 := H2⊖C, defined by Tnf(z) = f(zn). Although the semigroup Tn : n ≥ 1 and Wn : n ≥ 1

are not unitarily equivalent, they are semiconjugate; this is, Tn(I − S) = (I − S)Wn, where S is the shift of

multiplication by z in H2. This relation allowed S. Waleed Noor to guarantee that cyclic vectors of Wn : n ≥ 1are properly embedded into the PDCP functions.

The purpose of this work is to investigate this semigroup Wn : n ≥ 1. In particular, we introduce a new

reformulation of the Riemann hypothesis in terms of the invariance of the Hilbert subspace spanned by hk : k ≥ 2under W ∗

n , for any n ≥ 2. This result lead us to focus on the study of invariant subspaces of W ∗n : n ≥ 1. For

this reason, a series of questions will be discussed and we shall provide an answer. We also present a generalization

for the BA¡ez-Duarte criterion in H2 trough a family of cyclic vector for Wn : n ≥ 1. Recall that at this point

there is only one known cyclic vector: the constant 1 in H2(D).

2 Main Results

Theorem 2.1. Let N be the linear span of hk : k ≥ 2. Then the Riemann hypothesis is true if and only if the

closure of N is W ∗k -invariant for any k ≥ 2.

In order to generalize the BA¡ez-Duarte criterion in H2, we provide a family of cyclic vectors for Wn : n ≥ 1.

Let pm,λ(z) := zm + · · · + z − λ, m ∈ N and λ ∈ C.

Theorem 2.2. pm,λ is a cyclic vector for Wn : n ≥ 1, for every m ∈ N and λ ∈ C such that |λ+ 1| >√m+ 1.

Corollary 2.1. Let m ∈ N and λ ∈ C such that |λ + 1| >√m+ 1. Then the Riemann hypothesis is true if and

only if pm,λ belongs to the closure linear span of hk : k ≥ 2 in H2.

References

[1] bA¡ez-duarte, l. - A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis. Atti Acad.

Naz. Lincei, 14 (2003) 5-11.

[2] beurling, a. - A closure problem related to the Riemann zeta-function. Proc. Natl. Acad. Sci., 41 (1955)

312-314.

[3] nikolski, n. - In a shadow of the RH: cyclic vectors of the Hardy spaces on the Hilbert multidisc. Ann. Inst.

Fourier, 62(5), 1601-1626 (2012).

[4] nyman, b. - On some groups and semigroups of translations, Thesis, Uppsala, 1950.

[5] olofsson, a. - On the shift semigroup on the Hardy space of the Dirichlet series, Acta Math. Hungar., 128

(3) (2010), 265-286.

[6] rosenblum, m. and rovnyak, j. - Hardy classes and operator theory, Courier Corporation, 1997.

[7] waleed noor, s. - A Hardy space analysis of the BA¡ez-Duarte criterion for the RH. Adv. Math., 350 (2019)

242-255.


ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO NONLINEAR INTEGRAL EQUATIONS VIA

RENORMALIZATION

GASTAO A. BRAGA1, JUSSARA M. MOREIRA2 & CAMILA F. SOUZA3

1Universidade Federal de Minas Gerais, UFMG, MG, Brasil, [email protected],2Universidade Federal de Minas Gerais, UFMG, MG, Brasil, [email protected],

3Departamento de Matematica, Centro Federal de Educacao Tecnologica de Minas Gerais, MG, Brasil,

[email protected]

Abstract

In this work we use the renormalization group method to study the long-time asymptotics of solutions to

a class of nonlinear integral equations with a generalized heat kernel. We classify the nonlinearities according

to its role in the asymptotic behavior and we prove that, if one adds nonlinearities classified as irrelevant in

the renormalization group sense, then the behavior of the solution in the limit as t goes to infinity remains

unchanged but if marginal nonlinear terms are added in the equation, then the asymptotic behaviour acquires

an extra logarithmic decay factor.

1 Introduction

Here we use the renormalization group method as proposed by Bricmont et al. [1] to study nonlinear integral

equations with a generalized heat kernel, obtaining global existence and uniqueness of the solution, as well as the

asymptotic behavior. The nonlinearities are classified according to their contribution to the asymptotic behavior.

We show that the so called irrelevant perturbations do not affect the asymptotic profile of the solution, in the sense

that the profile is the same as in the linear case, whereas, by adding marginal perturbations, an extra logarithmic

factor appears on the decay of the solution.

Our proof relies on the Renormalization Group approach which was originally introduced in quantum field

theory and statistical mechanics and it was afterwards applied to the asymptotic analysis of deterministic differential

equations, both analytically and numerically. It proved to be very useful on the asymptotic analysis in problems

involving an infinite number of scales and has been used since then in different applications and approaches. Our

results here generalize the problems presented in [2, 3], where the Renormalization Group method was applied to

study the asymptotic behavior of the solution to I.V.P. ut = c(t)uxx + λF (u), t > 1, x ∈ R, u(x, 1) = f(x) with

c(t) = tp + o(tp) and nonlinearities of type F (u) =∑

j≥α ajuj .

2 Main Results

More specifically, we obtain the asymptotic behavior of solutions to equations of type

u(x, t) =

∫G(x− y, s(t))f(y)dy +

∫ t

1

∫G(x− y, s(t) − s(τ))F (u(y, τ))dydτ , (1)

with x ∈ R and t > 1. By imposing conditions on the kernel without specifying G = G(x, t), we generalize the study

of asymptotics for initial value problems. We consider therefore the integral kernel G(x, t) satisfying the following

general conditions:

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140

(i) There are integers q > 1 and M > 0 such that G(·, 1) ∈ Cq+1(R) and

supx∈R

(1 + |x|)M+2|G(j)(x, 1)| <∞, j = 0, 1, ..., q + 1,

where G(j)(x, 1) denotes the j-th derivative (∂jxG)(x, 1).

(ii) There is a positive constant d such that

G(x, t) = t−1dG(t−

1dx, 1

), x ∈ R, t > 0;

(iii) G(x, t) =∫RG(x− y, t− s)G(y, s)dy, for x ∈ R and t > s > 0.

This outlook was adopted in [4, 5] where it is shown that, under similar conditions on G, with s(t) = t, the

solution u(x, t) to (1) behaves for long time as

A

t1/dG( x

t1/d, 1),

where d > 0 is such that G(x, t) = t−1dG(t−

1dx, 1

). We recover and extend the above result using a renormalization

group approach showing that, if c(t) is a positive function in L1loc((1,+∞)) of type tp + o(tp), with p > 0 and

s(t) =

∫ t

1

c(τ)dτ =tp+1 − 1

p+ 1+ r(t), (2)

then, for F (u) =∑

j≥α ajuj with α > (p+ 1 + d)/(p+ 1),

u(x, t) ∼ A

t(p+1)/dG

(x

t(p+1)/d,

1

p+ 1

)when t→ ∞.

Furthermore, if F (u) = −µuαc + λ∑

j>α ajuj with µ small and positive and αc = (d+ p+ 1)/(p+ 1), then

u(x, t) ∼ A

(t ln t)(p+1)/dG

(x

t(p+1)/d,

1

p+ 1

)when t→ ∞.

References

[1] J. Bricmont, A. Kupiainen and G. Lin - Renormalization Group and Asymptotics of Solutions of Nonlinear

Parabolic Equations. Comm. Pure Appl. Math., 47, 893-922, 1994.

[2] G. A. Braga, F. Furtado, J. M. Moreira and L. T. Rolla - Renormalization Group Analysis of

Nonlinear Diffusion Equations with Time Dependent Coefficients: Analytical Results. Discrete and Continuous

Dynamical Systems. Series B, 7, 699–715, 2007.

[3] G. A. Braga and J. M. Moreira - Renormalization Group Analysis of Nonlinear Diffusion Equations with

Time Dependent Coefficients and Marginal Perturbations. Journal of Statistical Physics, 148, no. 2, 280–295,

2012.

[4] K. Ishige, T.Kawakami, and K.Kobayashi - Asymptotics for a Nonlinear Integral Equation with a

Generalized Heat Kernel. Journal of Evolution Equations, 14, 749–777, 2014.

[5] K. Ishige, T.Kawakami, and K.Kobayashi - Global Solutions for a Nonlinear Integral Equation with a

Generalized Heat Kernel. Discrete and Continuous Dynamical Systems S, 7, 767–783, 2014.

[6] G. A. Braga, J. M. Moreira and C. F. Souza - Asymptotics for Nonlinear Integral Equations with a

Generalized Heat Kernel using Renormalization Group Technique. J. Math. Phys., 60, 013507, 2019.


SHARP ESTIMATES FOR THE COVERING NUMBERS OF THE WEIERSTRASS FRACTAL

KERNEL

KARINA N. GONZALEZ1, DOUGLAS AZEVEDO2 & THAIS JORDAO3

1ICMC, Universidade de Sao Paulo, Brasil, [email protected],2DAMAT- Universidade Tecnologica Federal do Parana, Parana, Brasil,[email protected],

3ICMC, Universidade de Sao Paulo, Brasil,[email protected]

Abstract

In this job we use the infamous continuous and nowhere differentiable Weierstrass function as a prototype

to define a ‘Weierstrass fractal kernel’. We investigate properties of reproducing kernel Hilbert space (RKHS)

associated to this kernel by presenting an explicit characterization of this space. In particular, we show that this

space has a dense subset composed of continuous but nowhere differentiable functions. Moreover, we present

sharp estimates for the covering numbers of the unit ball of this space as a subset of the continuous functions.

1 Introduction

In 1872 K. Weierstrass presented a particular class of trigonometric series as a collection of continuous but nowhere

differentiable functions (CNDF). These functions are defined in terms of the following Fourier series expansion

wa,b(x) =

∞∑n=0

an cos(bnπ x), x ∈ R. (1)

For 0 < a < 1 it is clear that it defines a continuous bounded function. Under this assumption, Weierstrass

proved that wa,b is nowhere differentiable provided that ab ≥ 1 + 3π2 , with b an odd integer ([2, 3]). G.H. Hardy

relaxed this condition in [1] by showing that for ab ≥ 1, the Weierstrass function is nowhere differentiable.

Here, as usual, C([a, b]) stands for the normed real vector space of real-valued continuous functions on [a, b]

with the supremum norm.

Let I = [−1, 1]. We consider here W : I × I −→ R, the ‘Weierstrass Fractal kernel’, defined for 0 < a < 1, b an

integer such that ab ≥ 1, and given by

W (x, y) := wa,b(x− y), x, y ∈ I, (2)

where wa,b is the Weierstrass function (1). This is a continuous, nowhere differentiable, symmetric and positive

definite kernel. The theory of RKHS tell us that there exists only one RKHS HW := HW (I) having the Weierstrass

Fractal kernel as reproducing kernel. We present a complete characterization of the space HW , given in terms of

Fourier series expansions. It is expected that the functions in a RKHS inherit some of the properties of the

generating kernel such as smoothness.

If A is a subset of a metric space M and ϵ > 0, the covering number

C(ϵ, A) := C(ϵ, A,M)

is the minimal number of balls in M of radius ϵ covering the set A. Clearly, C(ϵ, A) <∞, whenever A is a compact

subset of M . For X,Y Banach spaces and T : X → Y an operator the covering numbers are defined in terms of unit

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142

balls as follows. For ϵ > 0, if BX and BY are the unit balls in X and Y , respectively, then the covering numbers of

T are

C(ϵ, T ) := C(ϵ, T (BX), Y ),

and given by

C(ϵ, T ) = min

n ∈ N : ∃y1, y2, ..., yn ∈ Y s.t. T (BX) ⊂n⋃

j=1

(yj + ϵBY )

.

Our main goal is to investigate the covering numbers of the embedding IW : HW → C(I), where HW is the

reproducing kernel Hilbert space associated to W defined in equation (2).

In this job we present upper and lower estimates for IW : HW → C(I) achieving tight bounds. In short, here,

the approach is mainly based on operator norm estimate of IW and some others related finite rank operators. In

[4] this approach was employed to obtain estimates for the covering numbers of the embedding operator over the

RKHS associated to the Gaussian kernel over non-empty interior sets of Rd.

Nevertheless, only for a few infinite-dimensional spaces there has been success in determining the precise

asymptotics of covering numbers.

2 Main Results

The main result of this job reads as follows.

Theorem 2.1. Let W as in (2). The covering numbers of the embedding IW : HW → C(I) behave asymptotically

as follows

ln(C(ϵ, IW )) ≍[ln(1/ϵ(1 − a)1/2

)]2ln (1/a)

, as ϵ→ 0+.

References

[1] Hardy, G.H. - Weierstrass‘s non-differentiable function.Trans. Amer. Math. Soc. 17, 301-325, 1916.

[2] Jarnicki, M., Pflug, P. - Continuous Nowhere Differentiable Functions: The Monsters of Analysis, Springer

Monographs in Mathematics, 2015.

[3] Johsen, J. - Simple Proofs of Nowhere-Differentiability for Weierstrass Function and Cases of Slow Growth.J

Fourier Anal Appl, 16, 17-33, 2010.

[4] Kuhn, T. - Covering numbers of Gaussian reproducing kernel Hilbert space. Jornal of complexity, 27, 489-499,

2011.


DIRICHLET SERIES WITH MAXIMAL BOHR’S STRIP

THIAGO R. ALVES1,†, LEONARDO S. BRITO2,‡ & DANIEL CARANDO3,§

1Departamento de Matematica, ICE, UFAM, AM, Brasil, 2Departamento de Matematica, ICE, UFAM, AM, Brasil,3Departamento de Matematica, Facultad de Cs. Exactas y Naturales, UBA, Buenos Aires, Argentina,

†[email protected], ‡[email protected], §[email protected]

Abstract

We study linear and algebraic structures in sets of Dirichlet series with maximal Bohr’s strip. More precisely,

we consider a set M of Dirichlet series which are uniformly continuous on the right half plane and whose strip

of uniform but not absolute convergence has maximal width, i.e., 1/2. We show that M contains an isometric

copy of ℓ1 (except zero) and is strongly ℵ0-algebrable. Also, we prove that there is a dense Gδ set such that any

of its elements generates a free algebra contained in M∪ 0.

1 Introduction

Mathematics is plenty of examples that seem to challenge the intuition. For instance, discontinuous additive

functions, Weierstrass’ Monsters, Peano curves, non-extendable holomorphic functions, and so on and so forth. The

counter-intuitiveness of these examples may lead us to believe they must be rare, but usually this is not the case.

Moreover, recent investigations are presenting a very interesting picture. Many of these peculiar examples/objects

are not only far away from being rare: in many situations, the set formed by these objects can even contain big

linear or algebraic structures. As a seminal example, Gurariy in [5] constructed infinite dimensional subspaces of

C([0, 1]) all whose nonzero elements are nowhere differentiable functions. Since then, a whole theory was built in

this direction, especially in the last years. Many of these advances are documented in the recent monograph [1].

Our main goal in the present work was to search for linear and algebraic structures in the set of Dirichlet series

with maximal Bohr’s strips (see below for the definition). A Dirichlet series is a series of the form∑∞

n=1 ann−s,

where the coefficients an are complex numbers and s is a complex variable. The natural domains of convergence

of Dirichlet series are half-planes. Given a Dirichlet series D =∑ann

−s we can consider three natural abscissas

which define the biggest half-planes on which D converges, converges uniformly and converges absolutely:

σc(D) ≤ σu(D) ≤ σa(D) .

It is not hard to see that

supD Dir. ser.

σa(D) − σc(D) = 1.

Harald Bohr was among the first to consider the problem of finding the maximal width of the strip on which a

Dirichlet series can converge uniformly but not absolutely (this strip is usually called Bohr’s strip). Thus, the so

called Bohr’s absolute convergence problem was to determine the number

S := supD Dir. ser.

σa(D) − σu(D) .

Bohr first showed in 1913 that S ≤ 1/2, and later in 1931 Bohnenblust and Hille proved that actually S = 1/2. For

a modern reference about the solution of this problem we refer to [4, Ch. 1-4].

For a ∈ R we let Ca denote the set of all complex numbers z such that Re z > a. Bohr’s fundamental theorem

(see [4, Theorem 1.13]) ensures that every bounded holomorphic function f : C0 → C which may be represented as

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144

a Dirichlet series in some half-plane converges uniformly on Cδ for each δ > 0. Let H∞ denote the space of all such

functions. It is well-known that H∞ is actually a Banach algebra when equipped with the supremum norm. As a

consequence of Bohr’s results, the absolute convergence problem and its solution can be written as

S = supD∈H∞

σa(D) =1

2. (1)

As expected, finding explicit Dirichlet series D ∈ H∞ such that σa(D) = 12 is not an easy task. However, in

this work we show that we have plenty of them and that the set of such series contains large linear and algebraic

structures. Moreover, we can get all this in a much smaller subalgebra of H∞ which we now define.

A Dirichlet polynomial is a Dirichlet series of the form∑N

n=1 ann−s. Let A(C0) denote the subalgebra of H∞

of all Dirichlet series which are uniform limits on C0 of a sequence of Dirichlet polynomials. Our goal is to study

the following set of Dirichlet series:

M :=D ∈ A(C0) : σa(D) =

1

2

.

Note that, by (1), the Dirichlet series belonging to M are those with maximal Bohr’s strip.

2 Main Results

Let E be a topological vector space and κ be a cardinal number. A subset Z ⊂ E is said to be κ-spaceable if

Z ∪0 contains a closed vector subspace of E with dimension κ. Moreover, we say that a subset Z ⊂ E is maximal

spaceable if Z is dim(E)-spaceable. Our first main result is the following.

Theorem 2.1. The set M∪ 0 contains an isometric copy of ℓ1. In particular, it is maximal spaceable.

Let us recall the precise definition of algebrability. Let X be an arbitrary set, A be an algebra of functions

f : X → C and κ be a cardinal number. A subset Z ⊂ A is said to be strongly κ-algebrable if there is a sub-algebra

B of A which is generated by an algebraically independent set of generators with cardinality κ and such that

B ⊂ Z ∪0. Let ℵ0 denote the cardinality of the natural numbers. With this, we can state our second main result.

Theorem 2.2. The set M is strongly ℵ0-algebrable. Also, there is a dense Gδ subset of A(C0) such that any of

its elements generates a free algebra contained in M∪ 0.

It is worth noting that some ideas for the proofs of our results come from [2, 3].

References

[1] aron, r. m., bernal-gonzalez, l., pellegrino, d. and seoane-sepulveda, j. b. - Lineability: The Search

for Linearity in Mathematics., Monographs and Research Notes in Mathematics, Chapman & Hall/CRC, Boca

Raton, FL, 2016.

[2] bayart, f. and quarta, l. - Algebras in sets of queer functions. Israel J. Math., 158, 285-296, 2007.

[3] conejero, j. a., seoane-sepulveda, j. b. and sevilla-peris, p. - Isomorphic copies of ℓ1 for m-

homogeneous non-analytic Bohnenblust-Hille polynomials. Math. Nachr., 290, 218-225, 2017.

[4] defant, a., garcıa, d., maestre, m. and sevilla-peris, p. - Dirichlet Series and Holomorphic Functions

in High Dimensions., New Mathematical Monographs 37, Cambridge University Press, Cambridge, 2019.

[5] gurariy, v. i. - Linear spaces composed of everywhere nondifferentiable functions. (Russian) C. R. Acad.

Bulgare Sci. 44, no. 5, 13-16, 1991.


EXTENSOES DE ARENS DE MULTIMORFISMOS EM ESPACOS DE RIESZ E RETICULADOS DE

BANACH

GERALDO BOTELHO1,† & LUIS A. GARCIA2,‡

1Universidade Federal de Uberlandia, UFU, Brasil, 2Universidade de Sao Paulo, USP, SP, Brasil

†[email protected], ‡[email protected]

Abstract

Provamos que todas as extensoes de Arens de multimorfismos de Riesz de posto finito tomando valores em

espacos de Riesz Arquimedianos coincidem e sao multimorfismo de Riesz. Consequencias para extensoes de

Aron-Berner de multimorfismos de Riesz em reticulados de Banach sao obtidas.

1 Introducao

Para um espaco de Riesz E, por E∼ denotamos o espaco de todos os funcionais lineares regulares em E e por

E∼n o espaco de todos os funcionais lineares regulares ordem contınuos. Sejam E1, . . . , Em, F espacos de Riesz e

A : E1 × · · · × Em −→ F um operador m−linear regular. Em [1] foi mostrado que se E1, E2, F sao Arquimedianos

e A : E1 × E2 −→ F e um bimorfismo de Riesz, entao a extensao de Arens A∗∗∗ : (E∼1 )∼n × (E∼

2 )∼n −→ (F∼)∼n e

tambem um bimorfismo de Riesz. Para reticulados de Banach tal resultado foi mostrado em [2].

Seguindo nessa linha, neste trabalho provamos que: (i) para cada permutacao σ ∈ Sm, se A e positivo/regular,

entao todas as extensoes de Arens ARσm(A) : E∼∼

1 × · · · × E∼∼m −→ F∼∼ sao positivas/regulares; (ii) quando F

e Arquimediano, todas as extensoes de Arens ARσm(A) coincidem e sao multimorfismos de Riesz sempre que A

e multimorfismo de Riesz de posto finito; (iii) para o caso vetorial mostramos que para todo A multimorfismo

de Riesz, |ARσm(A)(x′′1 , . . . , x

′′m)|(y′) = ARσ

m(A)(|x′′1 |, . . . , |x′′m|)(y′) para todos x′′1 ∈ E∼∼1 , . . . , x′′m ∈ E∼∼

m e todo

y′ ∈ F∼ homomorfismo de Riesz. Consequencias desses resultados para extensoes de Aron-Berner em reticulados

de Banach sao provadas.


Por Lr(E1, . . . , Em;F ) denotamos o espaco vetorial de todos os operadores m−lineares regulares. Considere a

seguinte notacao: dados espacos de Riesz E1, . . . , Em, uma permutacao σ ∈ Sm e k ∈ 1, . . . ,m, denotamos

E1, . . . ,σ(1)E, . . . ,σ(k−1)E, . . . , Em =

E1, . . . , Em se k = 1,

E1, . . . , Em nessa mesma ordem onde Eσ(1), . . . ,

Eσ(k−1) sao retirados para k = 2, . . . ,m.

Por exemplo, (E1, 2E,E3) = (E1, E3). O mesmo define-se para (x1, . . . ,σ(1) x, . . . ,σ(k−1) x, . . . , xm). E para

k = 1, . . . ,m − 1, denotamos (E1, . . . ,σ(1)E, . . . , σ(k)E, . . . , Em) = (E1, . . . , Em) nessa mesma ordem, onde os

espacos de Riesz Eσ(1), . . . , Eσ(k) sao retirados. O mesmo define-se para (x1, . . . ,σ(1) x, . . . ,σ(k) x, . . . , xm). Se

k = m, denotamos Lr(E1, . . . ,σ(1)E, . . . ,σ(k)E, . . . , Em;R) = R.Fixe k ∈ 1, . . . ,m e sejam σ ∈ Sm uma permutacao e A ∈ Lr(E1, . . . ,σ(1)E, . . . ,σ(k−1)E, . . . , Em). Para cada

xr ∈ Er, r ∈ 1, . . . , n \ σ(1), . . . , σ(k) defina, A(x1, . . . ,σ(1) x, . . . ,σ(k) x; •; . . . , xm) : Eσ(k) −→ R,

A(x1, . . . ,σ(1) x, . . . ,σ(k) x; •; . . . , xm)(xσ(k)) = A(x1, . . . ,σ(1) x, . . . ,σ(k−1) x, . . . , xm),

onde o ponto • esta na σ(k)-esima coordenada. Cada funcional definido acima e linear e regular.

145

146

Definicao 2.1. Sejam E1, . . . , Em, F espacos de Riesz. Um operador m-linear A : E1 × . . .× Em −→ F e dito:

(i) Positivo (em notacao A ≥ 0 ou A ≤ 0) se A(x1, . . . , xm) ∈ F+, para todos x1 ∈ E+1 , . . . , xm ∈ E+

m.

(ii) Regular se A pode ser escrito como a diferenca de dois operadores positivos.

(iii) Um multimorfismo de Riesz se |A(x1, . . . , xm)| = A(|x1|, . . . , |xm|) para todos x1 ∈ E1, . . . , xm ∈ Em.

Proposicao 2.1. Para σ ∈ Sm, k ∈ 1, . . . ,m e x′′σ(k) ∈ E∼∼σ(k), considere os operadores

x′′σ(k)σ

: Lr(E1, . . . ,σ(1)E, . . . ,σ(k−1)E, . . . , Em) −→ Lr(E1, . . . ,σ(1)E, . . . ,σ(k)E, . . . , Em),

x′′σ(k)σ(A)(x1, . . . ,σ(1) x, . . . ,σ(k) x, . . . , xm) = x′′σ(k)(A(x1, . . . ,σ(1) x, . . . ,σ(k) x; •; . . . , xm)).

Entao x′′σ(k)σe linear e regular. Alem disso, se 0 ≤ x′′σ(k) ∈ E∼∼

σ(k), entao os operadores x′′σ(k)σsao positivos.

Teorema 2.1. Para σ ∈ Sm e A ∈ Lr(E1, . . . , Em;F ), considere ARσm(A) : E∼∼

1 × · · · × E∼∼m −→ F∼∼ dado por

ARσm(A)(x′′1 , . . . , x

′′m)(y′) =

(x′′σ(m)

σ · · · x′′σ(1)

σ)(y′ A) para todo y′ ∈ F∼.

(a) Cada operador ARσm(A) e regular e estende A no sentido que ARσ

m(A) (JE1, . . . , JEm

) = JF A.(b) Se A ∈ Lr(E1, . . . , Em;F ) e positivo, entao cada ARσ

m(A) e positivo.

Teorema 2.2. Se F e um espaco de Riesz Arquimediano e A : E1 × · · · ×Em −→ F e um multimorfismo de Riesz

de posto finito, entao todas as extensoes de Arens de A, ARσm(A) : E∼∼

1 × · · · ×E∼∼m −→ F∼∼, σ ∈ Sm, coincidem

e sao multimorfismos de Riesz.

Proposicao 2.2. Se A ∈ Lr(E1, . . . , Em;F ) e um multimorfismo de Riesz e σ ∈ Sm. Entao para cada

homomorfismo de Riesz y′ ∈ F∼, JF∼(y′) ARσm(A) e um multimorfismo de Riesz e |ARσ

m(A)(x′′1 , . . . , x′′m)|(y′) =

ARσm(A)(|x′′1 |, . . . , |x′′m|)(y′), para todo x′′1 ∈ E∼∼

1 , . . . , x′′m ∈ E∼∼m .

Proposicao 2.3. Se A ∈ Lr(E1, . . . , Em;F ) e um multimorfismo de Riesz e σ ∈ Sm. Entao:

(a) y∗∗∗ ABσm(A) e um multimorfismo de Riesz para cada homomorfismo de Riesz w∗-contınuo y∗∗∗ ∈ F ∗∗∗.

(b) |ABσm(A)(x∗∗1 , . . . , x

∗∗m )|(y∗) = ABσ

m(A)(|x∗∗1 |, . . . , |x∗∗m |)(y∗), para todo x∗∗1 ∈ E∗∗1 , . . . , x∗∗m ∈ E∗∗

m e qualquer

y∗ ∈ spanφ ∈ F ∗ : φ e um homomorfismo de Riesz.

Corolario 2.1. Seja F um reticulado de Banach tal que F ∗ tem uma base de Schauder formada por homomorfismos

de Riesz. Entao todas as extensoes de Aron-Berner de qualquer multimorfismo de Riesz tomando valores em F sao

multimorfismo de Riesz.

Exemplo 2.1. O corolario anterior se aplica para os seguintes espacos: c0, ℓp, 1 < p <∞, F um espaco de Banach

com base de Schauder 1-incondicional que nao contem uma copia de ℓ1, F um espaco de Banach reflexivo com base

de Schauder 1-incondicional, o espaco original de Tsirelson’s T ∗ e seu dual T , o espaco de Schreier’s S, o predual

d∗(w, 1) do espaco de sequencias de Lorenz d(w, 1), cada reticulado de Banach F que e uma faixa projetada em

qualquer dos reticulados de Banach listados acima.

References

[1] Boulabiar, K., Buskes, G., Page, and R. - On Some Properties of Bilinear Maps of Order Bounded

Variation. Springer, Positivity 9, 401-414, 2005.

[2] Scheffold, E. - Ober die Arens-Triadjungierte Abbildung von Bimorphismen. Rev. Roumaine Math. Pures

Appl., 41, 697-701, 1996.


UNIVERSAL TOEPLITZ OPERATORS ON THE HARDY SPACE OVER THE POLYDISK

MARCOS S. FERREIRA1

1DCET, UESC, BA, Brasil, [email protected]

In collaboration with S. Waleed Noor and JoA£o R. Carmo

Abstract

The Invariant Subspace Problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-

trivial closed invariant subspace. Due to the existence of universal operators (in the sense Rota), the ISP may

be solved by describing the invariant subspaces of theses operators alone. We characterize all analytic Toeplitz

operators Tϕ on the Hardy space H2(Dn) over the polydisk Dn for n > 1 whose adjoints satisfy the Caradus

criterion for universality, that is, when T ∗ϕ is surjective and has infinite dimensional kernel. In particular, if ϕ

is a non-constant inner function on Dn, or a polynomial in the ring C[z1, · · · , zn] that has zeros in Dn but is

zero-free on Tn, then T ∗ϕ is universal for H2(Dn). The analogs of theses results for n = 1 are not true.

1 Introduction

One of the most important open problems in operator theory is the ISP, which asks: Given a complex separable

Hilbert space H and a bounded linear operator T on H, does T have a non-trivial invariant subspace? An invariant

subspace of T is a closed subspace E ⊂ H such that TE ⊂ E. The recent monograph by Chalendar and Partington

[2] is a reference for some modern approaches to the ISP. In 1960, Rota [7] demonstrated the existence of operators

that have an invariant subspace structure so rich that they could model every Hilbert space operator.

Definition 1.1. Let B be a Banach space and U a bounded linear operator on B. Then U is said to be universal

for B, if for any bounded linear operator T on B there exists a constant α = 0 and an invariant subspace M for U

such that the restriction U |M is similar to αT .

If U is universal for a separable, infinite dimensional Hilbert space H, then the ISP is equivalent to the assertion

that every minimal invariant subspace for U is one dimensional. The main tool thus far for identifying universal

operators has been the following criterion of Caradus [1].

Theorem 1.1. Let H be a separable infinite dimensional Hilbert space and U a bounded linear operator on H. If

ker(U) is infinite dimensional and U is surjective, then U is universal for H.

Let D be the unit disk in the complex plane C and T be the boundary of D. The polydisk Dn and torus Tn are

the cartesian products of n copies of D and T, respectively. We let Lp(Tn) = Lp(Tn, σ) denote the usual Lebesgue

space on Tn, where σ = σn is the normalized Haar measure on Tn, and L∞(Tn) the essentially bounded functions

with respect to σ. The Hardy space H2(Dn) is the Hilbert space of holomorphic functions f on Dn satisfying

∥f∥2 := sup0<r<1

∫Tn

|f(rζ)|2dσ(ζ) <∞.

Denote by H∞(Dn) the space of bounded analytic functions on Dn. It is well-known that both H2(Dn) and

H∞(Dn) can be viewed as subspaces of L2(Tn) and L∞(Tn) respectively by identifying f with its boundary function

f(ζ) := limr→1 f(rζ) for almost every ζ ∈ Tn. If |f | = 1 almost everywhere on Tn, then f is called inner function.

147

148

Let P denote the orthogonal projection of L2(Tn) onto H2(Dn). The Toeplitz operator Tϕ with symbol ϕ in

L∞(Tn) is defined by

Tϕf = P (ϕf)

for f ∈ H2(Dn). Just like on the disk, we have that Tϕ is a bounded linear operator on H2(Dn) and T ∗ϕ = Tϕ.

Moreover, if ϕ ∈ H∞(Dn), then Tϕf = ϕf for all ϕ ∈ H2(Dn) and Tϕ is called an analytic Toeplitz operator.

The best known examples of universal operators are all adjoints of analytic Toeplitz operators on H2(D), or

are equivalent to one of them. For example T ∗ϕ when ϕ is a singular inner function or infinite Blaschke product.

In the last few years, Cowen and Gallardo-Gutierrez [3, 4, 5, 6] have undertaken a thorough analysis of adjoints of

analytic Toeplitz operators that are universal for H2(D). The objective of this presentation is to consider analytic

Toeplitz operators Tϕ whose adjoints are universal on H2(Dn) for n > 1.

2 Main Results

Theorem 2.1. Let ϕ ∈ H∞(Dn) for n > 1. Then T ∗ϕ satisfies the Caradus criterion for universality if and only if

ϕ is invertible in L∞(Tn) but non-invertible in H∞(Dn).

Corollary 2.1. Let Tϕ be a left-invertible analytic Toeplitz operator on H2(Dn) for some n > 1. Then either Tϕ

is invertible or T ∗ϕ is universal.

References

[1] caradus, s. r. - Universal operators and invariant subspaces. Proc. Amer. Math. Soc. 23, 526-527, 1969.

[2] chalendar, i. and partington, j. r. - Modern approaches to the invariant subspace problem. Cambridge

University Press, 2011.

[3] cowen, c. c. and gallardo-gutierrez, e. a. - Consequences of universality among Toeplitz operators. J.

Math. Anal. Appl. 432, 484-503, 2015.

[4] cowen, c. c. and gallardo-gutierrez, e. a. - Rota’s universal operators and invariant subspaces in

Hilbert spaces. J. Funct. Anal. 271, 1130-1149, 2016.

[5] cowen, c. c. and gallardo-gutierrez, e. a. - A new proof of a Nordgren, Rosenthal and Wintrobe

theorem on universal operators. Problems and recent methods in operator theory, 97-102, Contemp. Math.,

687 Amer. Math. Soc., Providence, RI, 2017.

[6] cowen, c. c. and gallardo-gutierrez, e. a. - A hyperbolic universal operator commuting with a compact

operator. Proc. Amer. Math. Soc. (2020) https://doi.org/10.1090/proc/13922.

[7] rota, g. c. - On models for linear operators. Comm. Pure Appl. Math. 13, 469-472, 1960.


CICLICIDADE E HIPERCICLICIDADE DE OPERADORES DE COMPOSICAO NO ESPACO DE

HARDY DO SEMI-PLANO DIREITO

OSMAR R. SEVERIANO1

1Programa Associado de Pos-Graduacao em Matematica, UFPB/UFCG, PB, Brasil, [email protected]

Abstract

Seja C+ := z ∈ C : Re(z) > 0 o semi-plano direito. Neste trabalho, estudamos os operadores de composicao

CΦf = f Φ induzidos no espaco de Hardy do semi-plano direito H2(C+) por funcoes holomorfas Φ : C+ −→ C+.

Aqui caracterizamos completamente os operadores de composicao cıclicos e hipercıclicos em H2(C+) que sao

induzidos por funcoes da forma Φ(z) = az + b, onde a > 0 e Re(b) ≥ 0.

1 Introducao

Seja C+ := z ∈ C : Re(z) > 0 o semi-plano direito. O espaco de Hardy do semi-plano direito, denotado por

H2(C+), e o espaco de Hilbert de todas as funcoes holomorfas f : C+ −→ C para o qual

∥f∥ :=

(supx>0

1

2π

∫ ∞

−∞|f(x+ iy)|2dy

)1/2

(1)

e finito. A quantidade (1) descreve a norma de Hilbert de H2(C+).

Se Φ : C+ −→ C+ e uma funcao holomorfa, entao o operador de composicao CΦ com sımbolo Φ e definido por

CΦf = f Φ, f ∈ H2(C+).

A enfase na teoria de operadores de composicao esta na comparacao das propriedades de CΦ com as do sımbolo Φ.

Por exemplo, Elliott e Jury mostraram que CΦ e limitado em H2(C+) se, e somente se, Φ tem derivada angular

finita em ∞ (veja [1, Theorem 3.1]). Relembre que uma transformacao fracionaria linear de C+ e uma funcao

Φ : C+ −→ C+ da forma

Φ(z) =az + b

cz + d, z ∈ C+.

Devido ao criterio de limitacao para CΦ segue que as transformacoes fracionarias lineares de C+ que induzem

operadores de composicao limitados em H2(C+) tem a forma

Φ(z) = az + b, z ∈ C+ (2)

onde a > 0 e Re(b) ≥ 0.

Sejam X um espaco normado e L(X) o espaco de todos os operadores lineares limitados T : X −→ X. Um

operador T ∈ L(X) e cıclico se existe um vetor x ∈ X tal que o espaco gerador por Tnxn∈N e denso em X. Se

Tnxn∈N e denso em X, entao T e dito ser hipercıclico. Nestes casos, x e chamado de vetor cıclico e hipercıclico,

respectivamente. Aqui caracterizamos quais dos sımbolos em (2) induzem operadores de composicao cıclicos ou

hipercıclicos.


Os principais resultados deste trabalho podem ser sumarizados na seguinte tabela:

As demonstracoes dos resultados apresentados na tabela podem ser encontrados em [2].

149

150

Sımbolo Φ(z) = az + b Ciclicidade de CΦ Hiperciclicidade de CΦ

Re(b) = 0 Nao Nao

a = 1 and Re(b) > 0 Sim Nao

a < 1 and Re(b) > 0 Nao Nao

a > 1 and Re(b) > 0 Sim Nao

References

[1] elliot, s.j. and jury, m.t. - Composition operators on Hardy spaces of a half-plane, Bull. Lond. Math. Soc.

44 (3) (2012) 489-495.

[2] noor, s.w. and severiano, o. r. - Complex symmetry and cyclicity of composition operators on H2(C+),

Proc. Amer. Math. Soc. 148 (2020), 2469-2476.


ANEIS LINEARMENTE TOPOLOGIZADOS ESTRITAMENTE MINIMAIS

PATRICIA C. G. MAURO1 & DINAMERICO P. POMBO JR.2

1Centro Interdisciplinar de Ciencias da Natureza, UNILA, PR, Brasil, [email protected],2Instituto de Matematica e Estatıstica, UFF, RJ, Brasil, [email protected]

Abstract

Nesta nota introduzimos a nocao de anel linearmente topologizado estritamente minimal, provamos que todo

anel de valorizacao discreta e estritamente minimal e fornecemos condicoes necessarias e suficientes para que um

anel linearmente topologizado de Hausdorff seja estritamente minimal.

1 Introducao

Nesta nota anel significara anel comutativo com elemento unidade diferente de 0 e modulo significara modulo

unitario.

Definicao 1.1. Um anel linearmente topologizado [1] (§7) de Hausdorff (R, τR) (munido de sua estrutura canonica

de R-modulo) e dito estritamente minimal se toda topologia de Hausdorff em R que o torne um (R, τR)-modulo

linearmente topologizado coincidir com τR.

O resultado a seguir fornece exemplos importantes de aneis linearmente topologizados estritamente minimais.

Proposicao 1.1. Sejam R um anel de valorizacao discreta e τR sua topologia [4] (Capıtulo I). Entao (R, τR) e

estritamente minimal.

Prova: Com efeito, se τ e uma topologia de Hausdorff em R tal que (R, τ) e um (R, τR)-modulo linearmente

topologizado, da continuidade da aplicacao

(λ, µ) ∈ (R×R, τR × τ) 7−→ λµ ∈ (R, τ) (1)

segue a continuidade da aplicacao

λ ∈ (R, τR) 7−→ λ ∈ (R, τ); (2)

logo, τ e menos fina do que τR.

Reciprocamente, mostremos que τR e menos fina do que τ . De fato, sejam πR o ideal maximal de R e m

um inteiro ≥ 1 arbitrario. Como πm = 0 e τ e uma topologia de Hausdorff, existe uma τ -vizinhanca U de 0 em

R que e um ideal de R tal que πm /∈ U . Afirmamos que U ⊂ πmR, o que assegurara que τR e menos fina do

que τ . Realmente, seja v uma valorizacao discreta no corpo de fracoes K de R tal que R = λ ∈ K; v(λ) ≥ 0[4] (p. 17) e admitamos a existencia de ξ ∈ U tal que ξ /∈ πmR. Entao ξ = 0 e v(ξ) ∈ 0, 1, . . . ,m− 1.

Como 0 = v(1) = v(ξξ−1) = v(ξ) + v(ξ−1), v(ξ−1) = −v(ξ) ∈ −(m− 1), . . . ,−1, 0. Logo, ξ−1πm ∈ R, pois

v(ξ−1πm) = v(ξ−1) + v(πm) = v(ξ−1) +m > 0. Consequentemente, πm = ξ(ξ−1πm) ∈ UR ⊂ U , o que nao ocorre.

Portanto, U ⊂ πmR.

151

152


Argumentando como em [2, 3], podemos estabelecer o

Teorema 2.1. Para um anel linearmente topologizado de Hausdorff (R, τR), as seguintes condicoes sao equivalentes:

(a) (R, τR) e estritamente minimal;

(b) para todo (R, τR)-modulo linearmente topologizado de Hausdorff F , onde F e um R-modulo livre com uma base

de 1 elemento, todo isomorfismo de R-modulos de R em F e um homeomorfismo de (R, τR) em F ;

(c) todo R-modulo livre com uma base de 1 elemento admite uma unica topologia que o torna um (R, τR)-modulo

linearmente topologizado de Hausdorff;

(d) para todo (R, τR)-modulo linearmente topologizado E e para todo (R, τR)-modulo linearmente topologizado de

Hausdorff F , onde F e um R-modulo livre com uma base de 1 elemento, toda aplicacao R-linear sobrejetora de E

em F com nucleo fechado e contınua.

(e) para todo (R, τR)-modulo linearmente topologizado E e para todo (R, τR)-modulo linearmente topologizado de

Hausdorff F , onde F e um R-modulo livre com uma base de 1 elemento, toda aplicacao R-linear de E em F com

grafico fechado e contınua.

Como consequencia da Proposicao 1.1 e do Teorema 2.1 resulta que as condicoes (b), (c), (d) e (e) sao validas se

(R, τR) e um anel de valorizacao discreta arbitrario.

References

[1] Grothendieck, A. et Dieudonne, J. A. - Elements de Geometrie Algebrique I, Die Grundlehren der

mathematischen Wissenschaften 166, Springer-Verlag, Berlin - Heidelberg - New York, 1971.

[2] Nachbin, l. - On strictly minimal topological division rings. Bull. Amer. Math. Soc., 55, 1128-1136, 1949.

[3] pombo jr., d. p. - Topological modules over strictly minimal topological rings. Comment. Math. Univ.

Carolinae, 44, 461-467, 2003.

[4] serre, j. -p. - Corps locaux, Quatrieme edition, Actualites Scientifiques et Industrielles 1296, Hermann, Paris,

1968.


THE RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL AS A SEMIGROUP IN

BOCHNER-LEBESGUE SPACES

PAULO M. CARVALHO NETO1

1Departamento de Matematica, UFSC, SC, Brasil, [email protected]

Abstract

The Riemann-Liouville fractional integral is a classic tool from fractional calculus and the literature about

its properties is very huge. In this short communication, we would like to present this fractional integral as a

semigroup in L(Lp(t0, t1;X)), with respect to the order of integration, when t0, t1 ∈ R, with t0 < t1, and X is a

Banach space. Then we prove that its infinitesimal generator is an unbounded linear operator, which allows me

to conclude that the fractional integral is not an uniformly continuous semigroup.

1 Introduction

Let us begin by recalling the notions of fractional integral and Bochner-Lebesgue spaces Lp(t0, t1;X), when we have

t0, t1 ∈ R, with t0 < t1 and X a Banach space.

Definition 1.1. Consider 1 ≤ p ≤ ∞. We use the symbol Lp(t0, t1;X) to represent the set of all Bochner

measurable functions f : I → X for which ∥f∥X ∈ Lp(t0, t1;R), where Lp(t0, t1;R) stands for the classical Lebesgue

space. Moreover, Lp(t0, t1;X) is a Banach space when considered with the norm

∥f∥Lp(t0,t1;X) :=

[∫ t1

t0

∥f(s)∥pX ds

]1/p, if p ∈ [1,∞),

ess sups∈[t0,t1] ∥f(s)∥X , if p = ∞.

Definition 1.2. For α ∈ (0,∞) and f : [t0, t1] → X, the Riemann-Liouville (RL for short) fractional integral of

order α at t0 of a function f is defined by

Jαt0,tf(t) :=

1

Γ(α)

∫ t

t0

(t− s)α−1f(s) ds, (1)

for every t ∈ [t0, t1] such that integral (1) exists. Above Γ(z) denotes the classical Euler’s gamma function.

With these definitions, and by considering Riesz-Thorin interpolation theorem, we are able to prove that:

Theorem 1.1. Let α > 0, 1 ≤ p ≤ ∞ and f ∈ Lp(t0, t1;X). Then Jαt0,tf(t) is Bochner integrable and belongs to

Lp(t0, t1;X). Furthermore, it holds that

[∫ t1

t0

∥∥Jαt0,tf(t)

∥∥pXdt

]1/p≤[

(t1 − t0)α

Γ(α+ 1)

]∥f∥Lp(t0,t1;X). (2)

In other words, Jαt0,t is a bounded linear operator from Lp(t0, t1;X) into itself.

153

154

2 Main Results

The results presented above, together with the Abstract Semigroup Theory, are enough for us to present the

following results:

Theorem 2.1. Let 1 ≤ p ≤ ∞. Then the family Jαt0,t : α ≥ 0 defines a C0−semigroup in Lp(t0, t1;X).

Theorem 2.2. Let 1 ≤ p ≤ ∞ and assume that A : D(A) ⊂ Lp(t0, t1;X) → Lp(t0, t1;X) is the infinitesimal

generator of the C0−semigroup Jαt0,t : α ≥ 0 in Lp(t0, t1;X). Then f ∈ D(A) if, and only if,∫ t

t0

ln(t− s)f(s) ds

is absolutely continuous from [t0, t1] into X and its derivative belongs to Lp(t0, t1;X). Moreover, we have

Af(t) = −ψ(1)f(t) +d

dt

[∫ t

t0

ln(t− s)f(s) ds

], (1)

for almost every t ∈ [t0, t1], where ψ(t) denotes the digamma function.

Finally we present the main result of this short communication.

Theorem 2.3. Assume that 1 ≤ p ≤ ∞. If A : D(A) ⊂ Lp(t0, t1;X) → Lp(t0, t1;X) is the infinitesimal generator

of the C0−semigroup Jαt0,t : α ≥ 0 ⊂ L(Lp(t0, t1;X)), then A : D(A) ⊂ Lp(t0, t1;X) → Lp(t0, t1;X) is an

unbounded operator.

Proof. If p = ∞, x ∈ X, with ∥x∥X = 1, and we define ϕ ∈ L∞(t0, t1, X) by ϕ(t) = x, then ϕ ∈ D(A), when D(A)

is viewed as a domain in L∞(t0, t1;X). This implies that D(A) ⊊ L∞(t0, t1;X), i.e., A ∈ L(L∞(t0, t1;X)).

If 1 ≤ p <∞ and we consider x ∈ X, with ∥x∥X = 1, n ∈ N∗ and the sequence ϕn(t) = (t− t0)nx, then

limn→∞

∥Aϕn∥Lp(t0,t1;X)

∥ϕn∥Lp(t0,t1;X)= ∞,

and therefore A is an unbounded operator.

3 Acknowledgement

It is worth emphasising that these results can be found in [3, 4], which are recently submitted works that were done

together with Prof. Renato Fehlberg Junior.

References

[1] Hille, E. and Phillips, R. S. - Functional Analysis and Semi-groups, Publications Amer. Mathematical

Soc., Colloquium Publications, 1996.

[2] P. M. Carvalho-Neto and R. Fehlberg Junior - On the fractional version of Leibniz rule. Math. Nachr.,

293(4), 670-700, 2020.

[3] P. M. Carvalho-Neto and R. Fehlberg Junior - The Riemann-Liouville fractional integral in Bochner-

Lebesgue spaces I. to appear.

[4] P. M. Carvalho-Neto and R. Fehlberg Junior - The Riemann-Liouville fractional integral in Bochner-

Lebesgue spaces II. to appear.


A HIPER-TRANSFORMADA DE BOREL POLINOMIAL

GERALDO BOTELHO1 & RAQUEL WOOD2

1Faculdade de Matematica, UFU, MG, Brasil, [email protected],2IME, USP, SP, Brasil, [email protected]

Abstract

Desenvolvemos neste trabalho uma tecnica para representar funcionais lineares em espacos de polinomios

homogeneos contınuos entre espacos de Banach, a qual denominamos de hiper-transformada de Borel polinomial.

Como exemplo de aplicacao desta tecnica, representamos os funcionais lineares em espacos de polinomios

compactos como operadores lineares integrais.

1 Introducao

A transformada de Borel polinomial classica e uma ferramenta muito utilizada para representacao de funcionais

lineares em espacos de polinomios homogeneos e funcoes holomorfas. Dados n ∈ N e espacos de Banach E e F ,

denotaremos por P(nE;F ) o espaco de Banach dos polinomios n-homogeneos contınuos de E em F com a norma

usual. Escrevemos L(E;F ) no caso linear e P(nE) no caso em que F e o corpo dos escalares. Se Q(nE;F ) e um

subespaco de P(nE;F ) munido de uma norma completa ∥ · ∥Q, a transformada de Borel polinomial e o operador

βn : (Q(nE;F ), ∥ · ∥Q)∗ −→ P(nE∗;F ∗) , βn(ϕ)(x∗)(y) = ϕ ([x∗]n ⊗ y) , x∗ ∈ E∗, y ∈ F,

onde ([x∗]n ⊗ y) (x) = x∗(x)ny e um polinomio n-homogeneo e contınuo.

Embora frutıfera, essa tecnica possui a seguinte limitacao: para βn ser injetiva, combinacoes lineares de

polinomios do tipo [x∗]n⊗y, chamados de polinomios de tipo finito, devem formar um subespcao denso de Q(nE;F ).

Em razao dessa restricao, desenvolvemos uma nova tecnica de representacao que, a grosso modo, contempla

subespacos maiores de P(nE;F ). Chamamos esta variante da transformada de Borel polinomial classica de hiper-

transformada de Borel polinomial, denotada por Bn, a qual e a contrapartida polinomial da hiper-tranformada de

Borel desenvolvida em [1] para representar funcionais lineares em espacos de operadores multilineares. No caso

linear temos β1 = B1 =: B.

Sejam (I, ∥·∥I) um ideal de Banach de operadores lineares no sentido de [3] e (I P(nE;F ), ∥·∥IP) o respectivo

ideal composicao de polinomios homogeneos no sentido de [2]. Tendo em vista as equivalencias em [1, Theorem

2.7], e esperado que as seguintes condicoes sejam equivalentes para quaisquer espacos de Banach E e F :

(i) A transformada de Borel linear B representa funcionais lineares em (I(E;F ), ∥ · ∥I).

(ii) A hiper-transformada de Borel polinomial Bn representa funcionais lineares em (I P(nE;F ), ∥ · ∥IP) para

todo n ∈ N.

(iii) A hiper-transformada de Borel polinomial Bn representa funcionais lineares em (I P(nE;F ), ∥ · ∥IP) para

algum n ∈ N.

Como e usual, alguns procedimentos multilineares funcionam bem para polinomios, enquanto que outros nao.

Experimentaremos as duas situacoes neste trabalho. Por um lado, veremos que (i) e (ii) sao equivalentes e,

obviamente, implicam (iii); mas nao sabemos se as tres condicoes sao equivalentes. E temos razoes para conjecturar

que nao e verdade em geral.

155

156


Dados P ∈ P(nE) e y ∈ F , denotaremos por P ⊗ y o polinomio em P(nE;F ) definido por

(P ⊗ y)(x) = P (x)y, x ∈ E.

Combinacoes lineares de polinomios deste tipo sao chamados de polinomios de posto finito. A hiper-transformda

de Borel polinomial e o operador Bn definido no proximo resultado.

Teorema 2.1. Seja Q(nE;F ) um subespaco vetorial de P(nE;F ) munido com uma norma completa ∥·∥Q contendo

os polinomios de posto finito e tal que ∥P ⊗ y∥Q ≤ ∥P∥ · ∥y∥ para todos P ∈ P(nE;F ) e y ∈ F . Entao:

(a) A aplicacao

Bn : (Q(nE;F ), ∥ · ∥Q)∗ −→ L(P(nE);F ∗) , Bn(ϕ)(P )(y) = ϕ(P ⊗ y),

e um operador linear bem definido e ∥Bn∥ ≤ 1.

(b) Bn e injetivo se, e somente se, o subespaco dos polinomios de posto finito e ∥ · ∥Q-denso em Q(nE;F ).

Semi-ideais a esquerda de operadors lineares foram introduzidos em [1]. Por propriedade geometrica de espacos

de Banach entendemos uma propriedade que e invariante por isomorfismos isometricos.

Teorema 2.2. Sejam (I, ∥ · ∥I) um ideal de Banach de operadores lineares, α um semi-ideal de operadores a

esquerda e P1 e P2 propriedades geometricas de espacos de Banach. Suponha que para todos espacos de Banach E

e F tais que E∗ tem P1 e F tem P2, a transformada de Borel linear B : (I(E,F ), ∥ · ∥I)∗ −→ Lα(E∗;F ∗) seja um

isomorfismo isometrico. Entao, para todo n ∈ N e todos espacos de Banach E e F tais que P(nE) tem P1 e F tem

P2, a hiper-transformada de Borel polinomial

Bn : (I P(nE;F ), ∥ · ∥IP)∗ −→ Lα(P(nE);F ∗) , Bn(ϕ)(P )(y) = ϕ(P ⊗ y),

e um isomorfismo isometrico.

Conforme dissemos, conjecturamos que a recıproca do teorema acima nao e verdadeira. De toda forma, temos

a seguinte recıproca parcial:

Teorema 2.3. Sejam (I, ∥ · ∥I) um ideal de Banach, α um semi-ideal de operadores a esquerda e E e F espacos

de Banach. Suponha que exista n ∈ N tal que a hiper-transformada de Borel polinomial Bn : (I P(nE;F ), ∥ ·∥IP)∗ −→ Lα(P(nE);F ∗) seja um isomorfismo isometrico sobre sua imagem. Entao, a transformada de Borel

linear B : (I(E;F ), ∥ · ∥I)∗ −→ Lα(E∗;F ∗) tambem e um isomorfismo isometrico sobre sua imagem.

Como exemplo de aplicacao do Teorema 2.2, no proximo resultado usamos a hiper-transformada de Borel

polinomial para representar funcionais lineares no dual hiper-ideal fechado PK dos polinomios compactos (polinomios

que transformam conjuntos limitados em conjuntos relativamente compactos). Veremos que, na presenca da

propriedade da aproximacao, os funcionais em PK podem ser representados por operadores lineares integrais. O

ideal de Banach dos operadores lineares integrais sera denotado por J .

Teorema 2.4. Se P(nE) ou F tem a propriedade da aproximacao, entao a hiper-transformada de Borel polinomial

Bn : [PK(nE;F )]∗ −→ J (P(nE);F ∗) e um isomorfismo isometrico.

References

[1] Botelho G. and Wood R. - On the representation of linear functionals on hiper-ideals of multilinear

operators., Banach J. Math. Anal. 15 (2021), no. 1, Paper No. 25, 23 pp.

[2] Botelho G, Pellegrino D. and Rueda P - On composition ideals of multilinear mappings and homogeneous

polynomials, Publ. Res. Inst. Math. Sci. 43 (2007), no. 4, 1139–1155.

[3] A. Pietsch - Operator Ideals, North-Holland, 1980.


SOME PROPERTIES OF ALMOST SUMMING OPERATORS

RENATO MACEDO1 & JOEDSON SANTOS2

1Departamento de Matematica, UFPB, JP, Brasil, [email protected],2Departamento de Matematica, UFPB, JP, Brasil, [email protected]

Abstract

In this work we will present the scope of three important results in the linear theory of absolute summing

operators. The first one was obtained by Bu and Kranz in [3] and it asserts that a continuous linear operator

between Banach spaces takes almost unconditionally summable sequences into Cohen strongly q-summable

sequences for any q ≥ 2, whenever its adjoint is p-summing for some p ≥ 1. The second of them states that

p-summing operators with hilbertian domain are Cohen strongly q-summing operators (1 < p, q < ∞), this

result is due to Bu [2]. The third one is due to Kwapien [7] and it characterizes spaces isomorphic to a Hilbert

space using 2-summing operators. We will show that these results are maintained replacing the hypothesis of

the operator to be p-summing by almost summing.

1 Introduction

If 1 ≤ p <∞, we say that a linear operator u : X → Y is absolutely p-summing (or p-summing) if (u(xi))∞i=1 ∈ ℓp(Y )

whenever (xi)∞i=1 ∈ ℓwp (X). The class of absolutely p-summing linear operators from X to Y will be represented

by Πp (X,Y ) (see [6]). In [5] Cohen introduced a class of operators which characterizes the p∗-summing adjoint

operators. If 1 < p ≤ ∞, we say that a linear operator u from X to Y is Cohen strongly p-summing (or strongly p-

summing) if (u(xi))∞i=1 ∈ ℓp⟨Y ⟩ whenever (xi)

∞i=1 ∈ ℓp(X). The class of Cohen strongly p-summing linear operators

from X to Y will be denoted by Dp(X,Y ). According to [1], a linear operator u ∈ L (X;Y ) is called to be almost

p-summing, 1 ≤ p <∞, if there is a constant C ≥ 0 such that∫ 1

0

∥∥∥∥∥m∑i=1

ri(t)u(xi)

∥∥∥∥∥2

dt

12

≤ C · ∥(xi)mi=1∥w,p

for every m ∈ N and x1, ..., xm ∈ X, whose ri are the Rademacher functions. The class of all almost summing

operators from X to Y is denoted by Πal.s.p (X,Y ). When p = 2, these operators are simply called almost summing

and we write Πal.s instead of Πal.s.2 (see [6, Chapter 12]). By [6, Proposition 12.5],⋃

1≤p<∞

Πp(X,Y ) ⊆ Πal.s(X,Y ).

Using strong tools as Pietsch domination theorem and Khinchin and Kahane inequalities, the main result obtained

by Bu and Kranz in [3] was:

Theorem 1.1. [3, Theorem 1] Let X and Y be Banach spaces and u be a continuous linear operator from X to Y .

If u∗ is p-summing for some p ≥ 1, then for any q ≥ 2, u takes almost unconditionally summable sequences in X

into members of ℓq⟨Y ⟩.

Let X be a Hilbert space. Cohen in [4] has shown that

Π2 (X,Y ) ⊆ D2 (X,Y ) for all Banach space Y. (1)

In [2], Bu showed that (1) is valid with no restrictions of the parameters p, q ∈ (1,∞) instead of p = q = 2. Cohen

[4] also asked if (1) characterizes spaces isomorphic to a Hilbert space. Kwapien [7] proved that this question has

a positive answer. These important results are as follows:

157

158

Theorem 1.2. [2, Main Theorem] Let 1 < p, q <∞, and let X be a Hilbert space and Y be a Banach space. Then

Πp (X,Y ) ⊆ Dq (X,Y ) .

Theorem 1.3. [7] The following properties of Banach space X are equivalent.

(i) The space X is isomorphic to a Hilbert space.

(ii) For every Banach space Y , Π2(X,Y ) ⊆ D2(X,Y ).

The work is organized as follows: We will present our first result which is an improvement on the Bu and Kranz

[3] result through a simpler argument than the original. Afterward, we will extend the statement of the main result

of Bu in [2, Main Theorem]. Finally, we will show a Kwapien type theorem using almost summing operators to

characterize spaces isomorphic to a Hilbert space.

2 Main Results

Theorem 2.1. (Extension of the Bu-Kranz Theorem) Let X and Y be Banach spaces and u be a continuous linear

operator from X to Y . If u∗ is almost p∗-summing for some p ≥ 1, then u takes almost unconditionally summable

sequences in X into members of ℓp⟨Y ⟩.

Theorem 2.2. (Extension Bu’s Theorem) Let 2 ≤ p <∞, 1 < q ≤ ∞, X and Y be Banach spaces such that X is

an Lp∗-space. Then

Πal.s.p (X,Y ) ⊆ Dq (X,Y ) .

Theorem 2.3. (Extension Kwapien’s Theorem) The following properties of Banach space X are equivalent.

(i) The space X is isomorphic to a Hilbert space.

(ii) For every 1 < q ≤ ∞ and every Banach space Y , Πal.s(X,Y ) ⊆ Dq(X,Y ).

(iii) For every Banach space Y , Πal.s(X,Y ) ⊆ D2(X,Y ).

References

[1] G. Botelho, H.A. Braunss and H. Junek, Almost p-summing polynomials and multilinear mappings, Arch.

Math., 76 (2001), 109–118.

[2] Q. Bu, Some mapping properties of p-summing operators with hilbertian domain, Contemp. Math., 328 (2003),

145–149.

[3] Q. Bu and P. Kranz, Some mapping properties of p-summing adjoint operators, J. Math. Anal. Appl., 303

(2005), 585–590.

[4] J. S. Cohen, A characterization of inner-product spaces using absolutely 2-summing operators, Studia Math.,

38 (1970), 271–276.

[5] J. S. Cohen, Absolutely p-summing, p-nuclear operators and their conjugates, Math. Ann., 201 (1973), 177–200.

[6] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced

Mathematics, Cambridge University Press, 1995.

[7] S. Kwapien, A linear topological characterization of inner product space, Studia Math., 38 (1970). 277–278.


ON THE BISHOP-PHELPS-BOLLOBAS THEOREM FOR BILINEAR FORMS FOR FUNCTION

MODULE SPACES

THIAGO GRANDO1,†

1Departamento de Matematica, DEMAT/G, UNICENTRO, PR, Brasil, [email protected]

Abstract

In this talk we study a version of the Bishop-Phelps-Bollobas theorem called Bishop-Phelps-Bollobas property

for bilinear forms. Under appropriate conditions for a function module space X we prove that the pair (X,X)

satisfies the BPBp for bilinear forms.

1 Introduction

Let E and F be Banach spaces. The Bishop-Phelps-Bollobas property for operators (BPBp for operators) has been

defined in [1], is a version of the Bishop-Phelps-Bollobas Theorem and is related to the density of the set of norm

attaining operators in the space of all bounded linear operators between E and F . Over the years, other versions

of this theorem have appeared. In [3] the authors defined another version of this theorem called Bishop-Phelps-

Bollobas property for bilinear forms (BPBp for bilinear forms) and proved that this property fails for bilinear forms

on ℓ1× ℓ1. In [2] Acosta, Becerra-Guerrero, Garcıa and Maestre presented classes of spaces satisfying this property,

such as, when the domain space E is an uniformly convex Banach space, then for every Banach space F , the pair

(E,F ) satisfies the BPBp for bilinear forms. It is known that the BPBp for bilinear forms on E × F implies the

BPBp for operators, and the converse is no longer true. Considering a function module space X, Grando and

LourenA§o [4], presented conditions for X such that the pair (ℓ1, X) satisfies the BPBp for operators. In this note,

will be present some conditions to the function module space X such that the BPBp for bilinear forms is satisfied

for the pair (X,X).

2 Main Results

Definition 2.1. Let E and F be Banach spaces. We say that the pair (E,F ) has the Bishop-Phelps-Bollobas

property for operators (shortly BPBp for operators) if given ε > 0, there is η(ε) > 0 such that whenever T ∈ SL(E,F )

and x0 ∈ SE satisfy that ∥Tx0∥ > 1− η(ε), then there exist a point u0 ∈ SE and an operator S ∈ SL(E,F ) satisfying

the following conditions

∥Su0∥ = 1, ∥u0 − x0∥ < ε, and ∥S − T∥ < ε.

Definition 2.2. Let E and F be Banach spaces. We say that the pair (E,F ) has the Bishop-Phelps-Bollobas

property for bilinear forms (shortly BPBp for bilinear forms) if given ε > 0, there are η(ε) > 0 and β(ϵ) > 0 with

limt→0 β(t) = 0 such that for any A ∈ SL2(E×F ) and (x0, y0) ∈ SE × SF is such that that |A(x0, y0)| > 1 − η(ε),

then are B ∈ SL2(E×F ) and (u0, v0) ∈ SE × SF satisfying the following conditions

|B(u0, v0)| = 1, ∥u0 − x0∥ < β(ϵ), ∥v0 − y0∥ < β(ϵ) and ∥B −A∥ < ε.

Definition 2.3. Function Module is (the third coordinate of) a triple (K, (Xt)t∈K , X), where K is a nonempty

compact Hausdorff topological space, (Xt)t∈K a family of Banach spaces, and X a closed C(K)-submodule of the

C(K)-module∏∞

t∈K Xt (the ℓ∞-sum of the spaces Xt) such that the following conditions are satisfied:

159

160

1. For every x ∈ X, the function t→ ∥x(t)∥ from K to R is upper semi-continuous.

2. For every t ∈ K, we have Xt = x(t) : x ∈ X.

3. The set t ∈ K : Xt = 0 is dense in K.

Theorem 2.1. Let (K, (Xt)t∈K , X) be a function module space. Suppose that for every xt ∈ Xt there exists f ∈ X

such that f(t) = xt and ∥f ≤ xt∥ then the pair (X,X) satisfies the BPBp for bilinear forms.

References

[1] acosta, m. d., aron, r. m., garcıa, d. and maestre, m. The Bishop-Phelps-Bollobas theorem for

operators. J. Funct. Anal., 254 (11), 2780-2799, (2008).

[2] acosta, m. d., becerra-guerrero, j., garcıa, d. and maestre, m. The Bishop-Phelps-Bollobas theorem

fails for bilinear forms. Trans. Amer. Math. Soc., 365, 5911-5932, (2013).

[3] choi, y.s. and song, h. g. The Bishop-Phelps-Bollobas theorem fails for bilinear forms on ℓ1 × ℓ1. J. Math.

Anal. Appl., 360, 752-753, (2009).

[4] grando, t. and lourenco, m.l. On a function module space with the approximate hyperplane series

property. J. Aust. Math. Soc., 108 (3), 341-348, (2020).


EM DIRECAO A UM TEOREMA ESPECTRAL PARA SEMIGRUPOS CONVOLUTOS

ALDO PEREIRA1

1Departamento de Matematicas, Universidad de La Serena, Chile, [email protected]

Abstract

Seja k uma funcao localmente integravel em [0,∞[. Semigrupos k-convolutos sao operadores que incluem

semigrupos e semigrupos integrados como casos particulares, e e parte da solucao fraca de certas equacoes

diferenciais funcionais de primeira ordem. Neste trabalho, o objetivo principal e obter uma versao do Teorema

Espectral para o espectro de pontos, o espectro aproximado e o espectro residual, para um semigrupo k-convoluto.

1 Introducao

Seja A : D(A) ⊆ X → X um operador linear e fechado, definido em um espaco de Banach X, cujo domınio nao

e necessariamente denso. Nosso interesse esta focado no espectro de A, denotado por σ(A). Em particular, se A

e gerador de uma famılia resolvente R(t) e dado λ ∈ σ(A), o objetivo e determinar o que podemos dizer sobre os

elementos de σ(R(t)), ou seja, quais sao as condicoes que permitem obter

σ(R(t)) = r(λ) : λ ∈ σ(A), (1)

para uma determinada funcao r(·) que depende de R(t). E conhecido da teoria que a igualdade (4) e satisfeita para

os casos de semigrupo e semigrupo integrado, nas referencias [3] e [2] respectivamente.

Para estabelecer os principais resultados deste trabalho, consideramos diferentes partes do espectro de um

operador A, chamadas de espectro de pontos, aproximado e residual, definidos respectivamente por

σp(A) = λ ∈ C : ker(λI −A) = 0,

σa(A) = λ ∈ C : (λI −A) nao e injetivo, ou ran(λI −A) nao e fechado,

σr(A) = λ ∈ C : ker(λI −A) = 0 ou ran(λI −A) = X.

A particao do espectro fornecida acima e aplicada no seguinte contexto. Seja A um operador fechado e

k ∈ L1loc(R+) tal que k(0) = 0, consideramos aqui a seguinte versao do problema abstrato de primeira ordem:

u′(t) = Au(t) + k(t)x,

u(0) = 0,

onde x ∈ X, e cuja solucao e dada pelo chamado semigrupo k-convoluto gerado por A, apresentado nas referencias [1]

e [3]. Este semigrupo e uma famılia fortemente contınua R(t)t≥0 ⊂ B(X) que satisfaz as seguintes propriedades:

1. R(t)x ∈ D(A) e R(t)Ax = AR(t)x para todo x ∈ D(A) e t ≥ 0.

2.

∫ t

0

R(s)x ds ∈ D(A) para todo x ∈ X e t ≥ 0, e R(t)x =

∫ t

0

k(s)x ds+A

∫ t

0

R(s)x ds.

161

162


Teorema 2.1. Seja R(t)t≥0 um semigrupo k-convoluto com gerador A em um espaco de Banach X. Entao, temos

σ(R(t)) ∪ 0 ⊇∫ t

0

k(t− s)eλs ds : λ ∈ σ(A)

∪ 0,

e as seguintes inclusoes sao certas:

σp(R(t)) ∪ 0 ⊇∫ t

0

k(t− s)eλs ds : λ ∈ σp(A)

∪ 0,

σa(R(t)) ∪ 0 ⊇∫ t

0

k(t− s)eλs ds : λ ∈ σa(A)

∪ 0.

Alem disso, se A e densamente definido, entao

σr(R(t)) ∪ 0 ⊇∫ t

0

k(t− s)eλs ds : λ ∈ σr(A)

∪ 0.

Prova: Para mostrar a validade das inclusoes acima, veja [5], Teoremas 5.3, 5.5, 5.6 e 5.7.

Teorema 2.2. Seja R(t)t≥0 um semigrupo k-convoluto com gerador A, entao

σp(R(t)) ∪ 0 =

∫ t

0

k(t− s)eλs ds : λ ∈ σp(A)

∪ 0.

Teorema 2.3. Seja R(t)t≥0 um semigrupo k-convoluto gerado por um operador A que e tambem gerador de um

C0-semigrupo, entao

σa(R(t)) ∪ 0 =

∫ t

0

k(t− s)eλs ds : λ ∈ σa(A)

∪ 0,

σr(R(t)) ∪ 0 =

∫ t

0

k(t− s)eλs ds : λ ∈ σr(A)

∪ 0.

References

[1] i. cioranescu - Local convoluted semigroups, Evolution equations (Baton Rouge, LA, 1992), 107-122, Lecture

Notes in Pure and Appl. Math. 168, Dekker, New York, 1995.

[2] c. day - Spectral mapping theorem for integrated semigroups, Semigroup Forum 47 (1993), 359-372.

[3] k. j. engel, r. nagel - One-parameter semigroups for linear evolution equations, GTM 194, Springer-Verlag,

New York, 2000.

[4] m. kostic, s. pilipovic - Global convoluted semigroups, Math. Nachr. 280 (2007), no. 15, 1727-1743.

[5] c. lizama, h. prado - On duality and spectral properties of (a,k)-regularized resolvents, Proc. Roy. Soc.

Edimburgh Sect. A 139 (2009), no. 3, 505-517.


SOLUTIONS FOR FUNCTIONAL VOLTERRA–STIELTJES INTEGRAL EQUATIONS

ANNA CAROLINA LAFETA1

1Departamento de Matematica, UnB, DF, Brasil, [email protected]

Abstract

In this work, we introduce a class of equations called functional Volterra–Stieltjes integral equations. This

type of equations encompasses many other kinds of equations such as functional Volterra equations, functional

Volterra equations with impulses, functional Volterra delta integral equations on time scales, functional fractional

differential equations with and without impulses, among others. We present here results concerning local

existence, uniqueness and prolongation of solutions.

1 Introduction

This presentation is based on the work [3]. Here, we are interested in a more general formulation of functional

Volterra integral equations involving the so called Stieltjes integral given by x(t) = ϕ(0) +

∫ t

τ0

a(t, s)f(xs, s)dg(s), t ⩾ τ0,

xτ0 = ϕ,

(1)

where the integral in the right–hand side is understood in the sense of Henstock–Kurzweil–Stieltjes, τ0 ⩾ t0,

ϕ ∈ G([−r, 0],Rn) and we assume the following conditions on the functions f , a and g:

(A1) The function g : [t0, d) → R is nondecreasing and left–continuous on (t0, d).

(A2) The function a : [t0, d)2 → R is nondecreasing with respect to the first variable and regulated with respect to

the second variable.

(A3) The Henstock–Kurzweil–Stieltjes integral ∫ τ2

τ1

a(t, s)f(xs, s)dg(s)

exists for each compact interval [τ0, τ0 + σ] ⊂ [t0, d), all x ∈ G([τ0 − r, τ0 + σ],Rn), t ∈ [t0, d) and all

τ0 ⩽ τ1 ⩽ τ2 ⩽ τ0 + σ.

(A4) There exists a locally Henstock–Kurzweil–Stieltjes integrable function M : [t0, d) → R+ with respect to g such

that for each compact interval [τ0, τ0 + σ] ⊂ [t0, d), we have∥∥∥∥∥∥τ2∫

τ1

(c1a(τ2, s) + c2a(τ1, s))f(xs, s)dg(s)

∥∥∥∥∥∥ ⩽

τ2∫τ1

|c1a(τ2, s) + c2a(τ1, s)|M(s)dg(s),

for all x ∈ G([τ0 − r, τ0 + σ],Rn), all c1, c2 ∈ R and all τ0 ⩽ τ1 ⩽ τ2 ⩽ τ0 + σ.

(A5) There exists a locally regulated function L : [t0, d) → R+ such that for each compact interval [τ0, τ0 + σ] ⊂[t0, d), we have ∥∥∥∥∥∥

τ2∫τ1

a(τ2, s)[f(xs, s) − f(zs, s)]dg(s)

∥∥∥∥∥∥ ⩽

τ2∫τ1

|a(τ2, s)|L(s) ∥xs − zs∥∞ dg(s),

for all x, z ∈ G([τ0 − r, τ0 + σ],Rn), and all τ0 ⩽ τ1 ⩽ τ2 ⩽ τ0 + σ.

163

164

This type of equation also encompasses impulsive Volterra–Stieltjes integral equations and Volterra functional

∆-integral equations.

2 Main Results

Our main results are the following.

Theorem 2.1. Assume f : G([−r, 0],Rn) × [t0, d) → Rn satisfies conditions (A3), (A4) and (A5), a : [t0, d)2 → Rsatisfies condition (A2) and g : [t0, d) → R satisfies condition (A1). Then for all τ0 ∈ [t0, d) and all ϕ ∈G([−r, 0],Rn), there exists a σ > 0 and a unique solution x : [τ0 − r, τ0 + σ] → Rn of the initial value problem: x(t) = ϕ(0) +

∫ t

τ0

a(t, s)f(xs, s)dg(s)

xτ0 = ϕ.

(2)

In the next theorem, let conditions (B1)–(B5) be the same as conditions (A1)–(A5) but with d = +∞.

Theorem 2.2. Suppose f : G([−r, 0],Rn)×[t0,+∞) → Rn satisfies conditions (B3), (B4) and (B5), a : [t0,+∞)2 →R satisfies condition (B2) and g : [t0,+∞) → R satisfies condition (B1). Then, for every τ0 ⩾ t0 and

ϕ ∈ G([−r, 0],Rn), there exists a unique maximal solution x : I → Rn of the equation (1), where I is a nondegenerate

interval with τ0 − r = min I. Also, I = [τ0 − r, ω), with ω ⩽ +∞.

Moreover, besides presenting results that guarantee existence and uniqueness of local and maximal solutions,

we also present the correspondences between equation (1) and impulsive Volterra–Stieltjes integral equations and

Volterra functional ∆-integral equations.

References

[1] M. Federson, R. Grau and J. G. Mesquita, Prolongation of solutions of measure differential equations

and dynamical equations on time scales, Math. Nachr. 2018.

[2] D. Frankova, Regulated functions, Mathematica Bohemica. 116 (1) (1991), 20-59.

[3] R. Grau, A. C. Lafeta, J. G. Mesquita, Functional Volterra Stieltjes integral equations and applications,

submitted.

[4] G. Gripenberg, S.-O. Londen, O. Steffans, Volterra Integral and Functional Equations, in: Encyclo[edia

of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990.


A CONJECTURA DE BESSE, ESPACO VACUO ESTATICO E ESPACO σ2-SINGULAR

MARIA ANDRADE1

1Departamento de Matematica, UFS, SE, Brasil, [email protected]

Abstract

Chamamos metricas CPE (Critical Point Equation) os pontos crıticos do funcional da curvatura escalar total

restrito ao espaco de metricas com curvatura escalar constante de volume unitario. Neste trabalho, daremos uma

condicao necessaria e suficiente para que uma metrica crıtica seja Einstein em termos de espacos σ2-singulares.

Tal resultado melhora nosso entendimento sobre metricas CPE e a conjectura de Besse com um novo ponto de

vista geometrico. Alem disso, provamos que a condicao CPE pode ser trocada pela condicao de espaco vacuo

estatico para caracterizar as variedades de Einstein fechadas em termos de espacos σ2-singulares.

1 Introducao

Uma variedade Riemanniana (Mn, g) e dita ser Einstein se o tensor de Ricci e multiplo da metrica g, i.e., Ricg = λg,

onde λ : M → R, em particular se (Mn, g) e conexa, entao λ e constante. Em outras palavras, (Mn, g) e Einstein

se o traco do tensor

Ricg = Ricg −Rg

ng

e identicamente zero, onde Ricg e Rg sao as curvaturas de Ricci e escalar, respectivamente.

Sejam (Mn, g) uma variedade conexa, fechada de dimensao n ≥ 3, M o espaco das metricas Riemnannianas e

S2(M) o espaco dos 2-tensores simetricos em M. Fischer e Marsden, ver [3], consideraram a aplicacao da curvatura

escalar R : M → C∞(M) que associa a cada metrica g ∈ M sua curvatura escalar. Sejam γg a linearizacao da

aplicacao R e γ∗g a sua adjunta L2-formal, entao eles usaram que

γg(h) = −∆gtrgh+ δ2gh− ⟨Ricg, h⟩

e

γ∗gf = ∇2f − (∆f)g − fRicg,

onde δg = −divg, h ∈ S2(M), ∇2g e a Hessiana e ∆gtrgh e o Laplaciano do traco de h, no estudo da sobrejetividade

da aplicacao da curvatura escalar Rg, e ainda consideraram a equacao de vacuo estatica γ∗g (f) = 0.

Nas ultimas decadas, varias pesquisas tem sido feitas nestes espacos. O problema de classificacao e uma questao

fundamental, assim como os resultados de rigidez. O funcional de Einstein-Hilbert S : M → R e definido por:

S(g) =

∫M

Rgdvg. (1)

Em 1987 Besse conjecturou, ver [2], que os pontos crıticos do funcional da curvatura escalar total (1), restrito

a M1 = g ∈ M ;Rg ∈ C e volg(M) = 1, onde C = g ∈ M ; Rg e constante = ∅, precisam ser Eisntein. Mais

precisamente, a equacao de Euler-Lagrange da acao Hilbert-Einstein restrita a M1 pode ser escrita como a seguinte

equacao do ponto crıtico (CPE)

γ∗gf = ∇2gf − (∆gf)g − fRicg = Ricg.

165

166


Nesta secao, serao apresentados alguns resultados.

Teorema 2.1. Seja (Mn, g, f), n ≥ 3, uma metrica CPE com funcao potencial nao constante f . (Mn, g) e Eisntein

se, e somente se, f ∈ KerΛ∗g, onde Λg : S2(M) → C∞(M) e a linerarizacao da σ2-curvatura e Λ∗

g e a adjunta

L2-formal do operador, i.e., (Mn, g, f) e um espaco σ2-singular.

Uma consequencia e a seguinte

Corolario 2.1. Seja (Mn, g, f), n ≥ 3, uma metrica CPE com funcao potencial nao constante f . Se f ∈ KerΛ∗g,

entao (Mn, g) e isometrica a esfera redonda com raio r =

(n(n− 1)

Rg

)1/2

e f e uma autofuncao do Laplaciano

associada ao primeiro autovalorRg

n− 1em Sn(r). Alem disso, dim KerΛ∗

g = n+ 1 e

∫M

fdvg = 0.

Alem disso, provamos que se (Mn, g) e uma variedade Riemanniana fechada e kergΛ∗g ∩ kerγ∗g = 0, entao

(Mn, g) e uma variedade de Einstein. Portanto, ela e isometrica a esfera redonda Sn.

Teorema 2.2. Seja (Mn, g, f) um espaco vacuo estatico, onde (Mn, g) e uma variedade Riemannina fechada de

dimensao n ≥ 3. (Mn, g) e Einstein se, e somente se, o espaco (Mn, g, f) e σ2-singular. Se f e uma funcao nao

constante, entao (Mn, g) e isometrica a esfera redonda Sn, em outro caso (Mn, g) deve ser Ricci plana.

Vale comentar que a abordagem utilizada para provar estes resultados sao: obter a linearizacao de uma certa

aplicacao geometrica, depois calcular a adjunta L2-formal dessa linearizacao e entender o nucleo dessa adjunta. Os

detalhes podem ser vistos em [1].

References

[1] andrade, m. - On the interplay between CPE metrics, vacuum static spaces and σ2-singular space.

ArXiv:2002.03365 . Aceito em Archiv der Mathematik 2021.

[2] besse, a. - Einstein manifolds., Springer Science & Business Media, 2007.

[3] fischer, a. and marsden, j. - Deformations of the scalar curvature. Duke Mathematical Journal, 42, 519–

547, 1975.


EIGENVALUE PROBLEMS FOR FREDHOLM OPERATORS WITH SET-VALUED

PERTURBATIONS

PIERLUIGI BENEVIERI1 & ANTONIO IANNIZZOTTO2

1Instituto de Matematica e Estatıstica, USP, SP, Brasil, [email protected],2Department of Mathematics and Computer Science, University of Cagliari, [email protected]

Abstract

By means of a suitable degree theory, we prove persistence of eigenvalues and eigenvectors for set-valued

perturbations of a Fredholm linear operator. As a consequence, we prove existence of a bifurcation point for a

non-linear inclusion problem in abstract Banach spaces. Finally, we provide applications to differential inclusions.

1 Introduction

The present paper is devoted to the study of the following eigenvalue problem with a set-valued perturbation:Lx− λCx+ εϕ(x) ∋ 0

x ∈ ∂Ω.(1)

Here L : E → F is a Fredholm linear operator of index 0 between two real Banach spaces E and F such that

kerL = 0, C is another bounded linear operator, Ω is an open subset of E not necessarily bounded and containing

0, ϕ : Ω → 2F is a locally compact, upper semi-continuous (u.s.c. for short) set-valued map of CJ-type, and λ, ε ∈ Rare parameters.

Problem (1) can be seen as a set-valued perturbation of a linear eigenvalue problem (which is retrieved for ε = 0):Lx− λCx = 0

x ∈ ∂Ω.(2)

So, it is reasonable to expect that, under suitable assumptions, solutions of (1) appear in a neighborhood of the

eigenpairs (x, λ) of (2). In fact, we show that this is the case for the trivial eigenpairs (x, 0), provided dim(kerL)

is odd, the set Ω ∩ kerL is compact, and the following transversality condition holds:

imL+ C(kerL) = F. (3)

More precisely, we denote S0 = ∂Ω ∩ kerL the set of trivial solutions of (2). We prove that there exist a rectangle

R = [−a, a] × [−b, b] (a, b > 0) and c > 0 such that for all ε ∈ [−a, a] the set of real parameters λ ∈ [−b, b] for

which (1) admits a nontrivial solution x ∈ E with dist(x,S0) < c is nonempty and depends on ε by means of an

u.s.c. set-valued map. Similarly, for all ε ∈ [−a, a] the set of vectors x ∈ E with dist(x,S0) < c that solve (1) for

some λ ∈ [−b, b] is nonempty and depends on ε by means of an u.s.c. set-valued map. This is usually referred to as

a persistence result for eigenpairs. Using such persistence, we prove that S0 contains at least one bifurcation point,

i.e., a trivial solution x0 such that any neighborhood of x0 in E contains a nontrivial solution.

This type of investigation of nonlinear eigenvalue problems goes back to various papers in the last two decades.

Here we extend the study of the problem to the case of a set-valued perturbation. Such an extension requires a more

general degree theory for set-valued maps, which extends Brouwer’s degree for nonlinear maps on C1-manifolds.

167

168

Such a degree theory has been introduced in [1] and redefined in [2] by a precise notion of orientation for set-valued

perturbations of nonlinear Fredholm maps between Banach spaces.

Our abstract results find a natural application to differential inclusions. Here we consider an ordinary differential

inclusion with Neumann boundary conditions and an integral constraint:u′′ + u′ − λu+ εΦ(u) ∋ 0 in [0, 1]

u′(0) = u′(1) = 0

∥u∥1 = 1.

Here Φ(u) : [0, 1] → 2R is a set-valued map depending on u, to be chosen according to several requirements (three

different examples will be presented). We shall prove that the transversality condition (3) holds, and hence the

above problem admits at least one bifurcation point.

2 Main Results

Theorem 2.1. Let dim(kerL) be odd, (3) hold, and Ω0 be compact. Then, problem (1) has at least one bifurcation

point.

Proof We argue by contradiction: assume that S0 contains no bifurcation points, i.e., for all x ∈ S0 there exists

an open neighborhood Ux ⊂ E × R × R of (x, 0, 0), such that for all (x, ε, λ) ∈ S ∩ Ux we have (ε, λ) = (0, 0).

The family (Ux)x∈S0is an open covering of the compact set S0 × (0, 0) in E × R × R, so we can find a finite

sub-covering, which we relabel as (Ui)mi=1.

Let a, b, c > 0 be such that

Bc(S0) ×R ⊂m⋃i=1

Ui,

where as usual R = [−a, a] × [−b, b]. Thus, we have

S ∩(Bc(S0) ×R

)= S0 × (0, 0)

(i.e., there are no solutions in Bc(S0) × R except the trivial ones). By reducing a, b, c > 0 if necessary, for all

ε ∈ [−a, a] \ 0 there exist x ∈ ∂Ω ∩Bc(S0), λ ∈ [−b, b] such that (x, ε, λ) ∈ S, a contradiction.

References

[1] V. Obukhovskii, P. Zecca, V. Zvyagin, An oriented coincidence index for nonlinear Fredholm inclusions

with nonconvex valued perturbations, Abstr. Appl. Anal. 2006 (2006) Art. ID 51794, 21 pp.

[2] P. Benevieri, P. Zecca, Topological degree and atypical bifurcation results for a class of multivalued

perturbations of Fredholm maps in Banach spaces, Fixed Point Theory 18 (2017) 85–106.

[3] P. Benevieri, A. Iannizzotto, - Eigenvalue problems for Fredholm operators with set-valued perturbations,

Adv. Nonlinear Stud.,, 20, 3 (2020), 701–723.


ON A CLASS OF ABEL DIFFERENTIAL EQUATIONS OF THIRD KIND

SAMUEL NASCIMENTO CANDIDO1

1Department of Mechanical Engineering, UFF, RJ, Brazil, [email protected]

Abstract

Consider a class of Abel equations of third kind [d(x) + c(x)ym]y′x = a(x) + b(x)y. Suppose that each of

them has the same constants m ∈ R− −1 and the same real continuous functions a(x), then if there exists a

certain functional relation among the variable coefficients b(x), c(x) and d(x), we can construct new exact general

solutions that are shared by all these equations. This result, which improves and generalizes earlier results from

literature, is proved in the present work. Here the notation ·′x = d·dx

denotes the classical derivative with respect

to the independent variable x.

1 Introduction

The Abel nonlinear differential equations have been widely studied, either calculating their solutions (see [1, 2]),

or specifying their centers, or characterizing the behaviour of their solutions to obtain qualitative properties like

blow up or exponential decay in finite or infinity time (see [3]). Particularly, when it comes to calculating their

solutions, many authors search for functional relations between the variable coefficients and integrating factors that

allow the construction of exact analytic solutions (see [1, 2]).

In 2020, by means of Poincare compactification, Regilene Oliveira and Claudia Valls [3] classified the topological

phase portraits of the Abel equation of third kind

C(x)y2y′x = A(x) +B(x)y (1)

(where the functions A(x), B(x) and C(x) are polynomials in x) to understand the behaviour of their solutions.

This problem becomes very hard when the number of parameters in the equation increases and we know that

the analysis of particular solutions for the differential equations is very important for understanding the solutions

sets of a differential equation and for assisting qualitative and numerical studies. Thus, to collaborate with future

qualitative and numerical studies about cases with more parameters, we present a new theorem whose constructive

demonstration leads to exact general solutions for the following more general case of equation (1)

[d(x) + c(x)ym]y′x = a(x) + b(x)y. (2)

satisfying y = y(x), c(x), d(x) ∈ C1(x1, x2) and a(x), b(x) ∈ C(x1, x2), where x1, x2 ∈ R.

In fact, a new direct analytic method is introduced to obtain these solutions for the general form of equation (2)

with b(x), c(x), d(x) = 0 and a(x) can be equals to 0 or not. For this, we propose a new functional relation between

variable coefficients of equation (2) and we use an argument of integrating factors.

2 Main Result

In this section, we prove the following result:

Theorem 2.1. For the general form of the Abel equation of third kind (2) with b(x), c(x), d(x) = 0, if their variable

coefficients satisfy the functional relation

c′xd = c (b+ d′x) (1)

169

170

then equation (2) admits the exact implicit general solution

ym+1 +(m+ 1) d(x)

c(x)

y − 1

µ(x)

[∫a(x)

d(x)µ(x) dx+ C

]= 0, (2)

where µ(x) = exp[−∫ b(x)

d(x) dx]is an integrating factor and C is an arbitrary constant of integration.

Proof The proof is resumed as follows: firstly, equation (2) can be rewritten in the form

y′x +c(x)

d(x)ymy′x =

a(x)

d(x)+b(x)

d(x)y. (3)

By using differentiation rules, we deduce

c(x)

d(x)ymy′x =

1

m+ 1

[c(x)

d(x)ym+1

]′x

−[c(x)

d(x)

]′x

ym+1

.

If we insert the last equation in equation (3), we have[y +

c(x)

(m+ 1)d(x)ym+1

]′x

=a(x)

d(x)+b(x)

d(x)

y +

d(x)

b(x)

[c(x)

(m+ 1)d(x)

]′x

ym+1

.

Now, if we consider the functional relation between the variable coefficients

d(x)

b(x)

[c(x)

(m+ 1)d(x)

]′x

=c(x)

(m+ 1)d(x)⇒ c′xd = c (b+ d′x) ,

so we can assume

ψ = ψ(x) = y +c(x)

(m+ 1)d(x)ym+1 (4)

such that we obtain the linear differential equation

ψ′x − b(x)

d(x)ψ =

a(x)

d(x). (5)

Multiplying both sides of equation (5) by the integrating factor µ(x), we get

(µψ)′x = µ(x)

a(x)

d(x)⇒ ψ(x) =

1

µ(x)

[∫a(x)

d(x)µ(x) dx+ C

].

Therefore, we use relation (4) for returning to the original dependent variable y = y(x), so we obtain equation (2).

This completes the resuming proof of the Theorem. In other words, the Theorem says that if each element of an

equations set (2) satisfies relation (1) and has the same constants m ∈ R − −1 and the same real continuous

functions a(x), then all these elements (equations) have the same exact general solutions given by equation (2).

References

[1] l. bougoffa - Further Solutions Of The General Abel Equation Of The Second Kind: Use Of Julia’s Condition.

Applied Mathematics E-Notes, 14, 53-56, 2014.

[2] m. p. markakis - Closed-form solutions of certain Abel equations of the first kind. Applied Mathematics

Letters, 22, 1401-1405, 2009.

[3] regilene oliveira and clA¡udia valls - ON THE ABEL DIFFERENTIAL EQUATIONS OF THIRD

KIND. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS SERIES B, 25, 1821-1834, 2020.


CONVERGENCE OF ALEVEL-SET ALGORITHM FOR SCALAR CONSERVATION LAWS

ANIBAL. CORONEL1

1Departamento de Ciencias Basicas, Universidad del Bıo-Bıo, Chillan, Chile, [email protected]

Abstract

In this paper we study the convergence of the level-set algorithm introduced by Aslam for tracking the

discontinuities in scalar conservation laws in the case of linear or strictly convex flux function (2001, J. Comput.

Phy. 167, 413-438). The numerical method is deduced by the level-set representation of the entropy solution:

the zero of a level-set function is used as an indicator of the discontinuity curves and two auxiliary states, which

are assumed continuous through the discontinuities, are introduced. Following the ideas of (2015 Numer. Meth.

for PDE 31, 1310–1343), we rewrite the numerical level-set algorithm as a procedure consisting of three big

steps: (a) initialization, (b) evolution and (c) reconstruction. In (a) we choose an entropy admissible level-set

representation of the initial condition. In (b), for each iteration step, we solve an uncoupled system of three

equations and select the entropy admissible level-set representation of the solution profile at the end of the time

iteration. In (c) we reconstruct the entropy solution by using the level-set representation. Assuming that in the

step (b) we can use a monotone scheme to approximate each equation we prove the convergence of the numerical

solution of the level set algorithm to the entropy solution in Lp for p > 1. In addition, some numerical examples

focused on the elementary wave interaction are presented.

1 Introduction

In this work we introduce a convergent numerical method for the Cauchy problem for a scalar conservation law:

ut + (f(u))x = 0 for (x, t) ∈ QT := R× R+ with u(x, 0) = u0(x) for x ∈ R,

where f : R → R is the flux function and u is the conserved variable. We consider the following data assumptions:

u0 ∈ L∞(R) and f(u) = au (a constant) or f ∈ C2(R,R) and f ′′(u) ≥ α > 0, for all u ∈ R and some α > 0. We

recall that the weak solutions satisfy the jump-entropy conditions

[u]s = [f(u)], s =dX

dt, [u] = ul − ur, [f(u)] = f(ul) − f(ur), f ′(ul) > s(t) > f ′(ur),

through a discontinuity of u parameterized by (X(t), t).

sgn In order to introduce the numerical method, we consider some notation: sgn+(x) = 1IR+(x) and

sgn−(x) = −sgn+(−x), where 1IA : X → 0, 1 is defined by 1IA(x) = 1 for x ∈ A and 1IA(x) = 0 for x ∈ X − A;

a+ = maxa, 0 and a− = mina, 0; Pnj = sgn+((pnj+1 − pnj−1)/2∆x) and En

j =[1− sgn−

(f ′(unL,j)− f ′(unR,j)

)]/2,

for (j, n) ∈ Z× N. The numerical method called LS-scheme consist in three big steps:

(I) Initialization step. We consider a continuous function p0 : R → R such that it vanishes over the control

volumes where the function u0 is discontinuous, and we can make the calculus of entropy admissible states w0 and

v0 and the initial speed s0;

(II) Evolution step. The evolution considers three intermediate states: (a) The intermediate states wn+1/2

and vn+1/2 are calculated by applying a monotone scheme with numerical flux g, i.e. wn+1/2j = wn

j −λ(g(wn

j , wnj+1)−

g(wnj−1, w

nj ))

and vn+1/2j = vnj − λ

(g(vnj , v

nj+1) − g(vnj−1, v

nj ))

; (b) The level set equation state pn+1 is calculated

by : pn+1j = pnj − λ(snj )+(pnj − pnj−1)− λ(snj )−(pnj+1 − pnj ); and (c) Using the notation Pn

j , we introduce the discrete

171

172

left-right states and the extended discrete shock speed as follows: un+1L,j = Pn+1

j wn+1/2j + (1 − Pn+1

j ) vn+1/2j ,

un+1R,j = (1 − Pn+1

j ) wn+1/2j + Pn+1

j vn+1/2j and sn+1

j =f(un+1

L,j )−f(un+1R,j )

un+1L,j −un+1

R,j

; and (d) using the indicator Enj ,

we introduce the states wn+1 and vn+1 such that un+1 is consistent with the entropy condition: wn+1j =

wn+1/2j +

(vn+1/2j − w

n+1/2j

)(1 − sgn+(pnj )

)Enj and vn+1

j = vn+1/2j +

(w

n+1/2j − v

n+1/2j

)sgn+(pnj )En

j ; and

(III) Reconstruction step. In this step we apply the definition of level set representation to reconstruct un+1

from pn+1,wn+1 and vn+1, i.e. un+1j = sgn+(pn+1

j ) wn+1j +

(1 − sgn+(pn+1

j ))vn+1j .

2 Main Result

Theorem 2.1. Consider the assumptions: (A1) f satisfies the hypothesis of strict convexity or linearity; (A2) u0

satisfies the hypothesis of L∞ boundness; (A3) p0, v0 and w0 satisfy the requirements specified by the initialization

step; (A4) g is a monotone flux; (A5) The functions w∆, v∆, p∆ and u∆ defined from R×R+0 are determined by the

the LS-scheme; and (A6) λ satisfies the CFL condition λ∥f ′∥L∞([um,uM ]) < 1−ξ with ξ ∈]0, 1[. Then, the numerical

solution u∆ converges to u, the entropy solution of the Cauchy problem, in the strong topology of Lploc(R×R+

0 ) for

all p > 1 when ∆x→ 0.

References

[1] T. D. Aslam, A level set algorithm for tracking discontinuities in hyperbolic conservation laws I: scalar

equations, J. Comput. Phys. 167(2), 413–438, 2001.

[2] A. Coronel, P. Cumsille and M. Sepulveda. Convergence of a level-set algorithm in scalar conservation

laws. Numerical Methods Partial Differential Equations, 31(4), 1310–1343, 2015.

[3] S. Konyagin, B. Popov, and O. Trifonov. On convergence of minmod-type schemes. SIAM J. Numer.

Anal. 42 (5), 1978–1997, 2005.

[4] B. Popov and O. Trifonov. One-sided stability and convergence of the Nessyahu-Tadmor scheme. Numer.

Math. 104 (4), 539–559, 2006.

[5] B. Popov and O. Trifonov. Order of convergence of second order schemes based on the minmod limiter.

Math. Comp. 75(256), 1735–1753, 2006.


LAGRANGIAN–EULERIAN SCHEME FOR GENERAL BALANCE LAWS

EDUARDO ABREU1, EDUARDO P. BARROS2 & WANDERSON LAMBERT3

1IMECC, UNICAMP, SP, Brasil, [email protected],2 sem endereco,

3UNIFAL, Pocos de Caldas, MG, Brazil, [email protected]

Abstract

We present an extension of the Lagrangian-Eulerian method for solving general balance laws [2], by taking

into account a nonlinear accumulation term and a non-conservative source. This approach generalizes the no-

flow curves as in [1] and does not require the use of approximate/exact Riemann solvers. The scheme is easy to

implement and provides fast, accurate and stable results. We present fully/semi-discrete methods and illustrate

the robustness of the approach with numerical examples for the nontrivial Baer-Nunziato system [3].

1 No-flow fully/semi-discrete schemes for general balance laws

The Fully Discrete Lagrangian Eulerian Scheme (FDLE). Consider the following system

[A(u)]t + [F (u)]x = G(u) + [S(u)]x +N(u)[B(u)]x, x ∈ R, t > 0; u(x, 0) = η(x), u(x, t) : R× R+ −→ Rm. (1)

After the discretization, we introduce the control volumes Dnj = (t, x)/tn ≤ t ≤ tn+1, φn

j (t) ≤ x ≤ φnj+1(t), where

dφnj (t)

dt=F (u)

A(u), tn < t ≤ tn+1; φn

j (tn) = xj . (2)

The no-flow curves of the control volumes Dnj are determined by the IVPs (2), which play an important role in the

Lagrangian-Eulerian method. For numerical purposes, the solution of (2) can be approximate with a simple and

robust first-order linearization, which gives us φnj (t) = xnj + (t − tn)fnj , where fnj := F (u)

A(u) . For the well-posedness

of (2), we essentially require F (u)A(u) as Lipschitz. In case of blow-up singularities in the term F (u)

A(u) , for real-world

applications [2], we apply a flux-split modeling strategy [1], whenever necessary to naturally handle this situation.

Thus, by writing (1) in its divergence form and integrating over the control volume Dnj , we are able to explore the

properties of the no-flow curves through the divergent theorem (where Inj =∫Dn

j

[G(u) + ∂S(u)

∂x +N(u)∂B(u)∂x

]dx):

∫∫Dn

j

∇xt ·

[F (u)

A(u)

]dx=

∫ φnj+1(t

n+1)

φnj (t

n+1)

A(u(x, tn+1)) dx−∫ xn

j+1

xnj

A(u(x, tn)) dx = Inj , where Anj :=

1

∆x

∫Dn

j

A(u)dx (3)

is the cell averages and projecting back the results of (3) to the original discrete lattice in Dnj , and approximating

the integrals Inj with a simple and robust quadrature, we get the fully-discrete Lagrangian Eulerian scheme:

An+1j =

Anj−1 + 2An

j +Anj+1

4− ∆tn

4

[fnj + fnj+1

∆xn+1j

(Anj +An

j+1) −fnj−1 + fnj

∆xn+1j−1

(Anj−1 +An

j )

]+

+1

∆x

[(∆x

2+ fnj ∆tn

)Inj−1

∆xn+1j−1

+

(∆x

2− fnj ∆tn

)Inj

∆xn+1j

], (4)

173

174

where ∆xn+1j = ∆x+ (fnj+1 − fnj )∆t. In order to recover the cell averages approximation of the original variables

at each time step Unj := 1

∆x

∫Dn

j

u(x, t) dx we still need to solve the typically non-linear system A(Unj ) = An

j .

Semi Discrete Lagrangian Eulerian Scheme (SDLE). Starting from (4) we can write

An+1j = An

j − ∆t

∆x

[Fnj − Fn

j−1 +

(∆x

2+ fnj ∆t

) Inj−1

∆xn+1j−1

+

(∆x

2− fnj ∆t

) Inj

∆xn+1j

], (5)

where Fnk = 1

4

[∆x∆t (An

k −Ank+1) + ∆x

(fnk +fn

k+1)

∆xn+1k

(Ank +An

k+1)]; In

k =Ink

∆t and k ∈ j, j + 1. We stress that

Inj = O(∆t∆x), so Inj = O(∆x). Applying t → 0, the derivative

dAj(t)

dt= lim

t−→0

An+1j −An

j

∆tcan be replaced

in (5) and due to the no-flow property[∆x∆t

]∝ F (u)

A(u) , we can remove the blow-up singularity replacing ∆x∆t in (5) by

a stability condition that depends on F (u)A(u) , leading us to the semi-discrete Lagrangian–Eulerian scheme for balance

laws:

dAj(t)

dt= − 1

∆x

[Fn

j −Fnj−1 +

Inj + In

j−1

2

], (6)

where Fnj =

1

4[bj+ 1

2(An

j −Anj+1) + (fnj + fnj )(An

j +Anj+1)] and bj+ 1

2= max

j|fnj + fnj+1|.

Numerical Experiments. To illustrate the robustness of the methods FDLE and SDLE, we applied them

to solve the Baer-Nunziato system from [3], which models a two-phase reactive flow in detonation systems:

[α]t = −u[α]x + F + Cρ

[α ρ]t + [α ρ u]x = C[α ρ u]t + [α(ρ u2 + p)]x = p [α]x + M[α ρ E]t + [α u(ρ E + p)]x = u p [α]x + −pF + E

[αρ]t + [αρu]x = −C[αρu]t + [α(ρ u2 + p)]x = −p [α]x + −M[αρE]t + [α u(ρ E + p)]x = −u p [α]x + pF − E

. (7)

The variables are α, ρ, p, u, the volume fraction, density, pressure and velocity of the gas phase as well as α, ρ, p,

u are these same quantities for the solid phase. The solutions are shown in Figure 1 and follow the full model and

initial data as described in [3], with a very good agreement given by the methods FDLE and SDLE in coarse grids

by comparing our numerical results along with the available exact solution in [3].

0 0.5 10.75

0.8

0.85

0.9FDLE

Exact

0 0.5 10.75

0.8

0.85

0.9SDLE

Exact

Figure 1: Solutions for α with 128 points at time tM = 1: FDLE (left) and SDLE (right).

References

[1] Abreu, E. and Perez, J., A fast, robust, and simple Lagrangian-Eulerian solver for balance laws and

applications., Computers & Mathematics with Applications 77(9) (2019) 2310-2336.

[2] Abreu, E., Bustos, A. and Lambert, W., AAsymptotic behavior of a solution of relaxation system for

flow in porous media, XVI Int. Conf. Hyperbolic Problems: Theory, Numerics, Applications. Springer, 2016.

[3] Hennessey, M., Kapila, A. K. and Schwendeman, D. W., An HLLC-type Riemann solver and high-

resolution Godunov method for a two-phase model of reactive flow with general equations of state. Journal of

Computational Physics 405 (2020) 109180.


A POSITIVE LAGRANGIAN-EULERIAN SCHEME FOR HYPERBOLIC SYSTEMS

EDUARDO ABREU1, JEAN FRANCOIS2, WANDERSON LAMBERT3 & JOHN PEREZ4

1IMECC, UNICAMP, SP, Brasil, [email protected],2sem endereø,

3UNIFAL, Pocos de Caldas, MG, Brazil, [email protected],4 Instituto Tecnologico Metropolitano, Colombia, [email protected]

Abstract

In this work, we design a class of positivity preserving Semi-Discrete Lagrangian-Eulerian schemes for solving

multidimensional initial value problems for scalar models and systems of conservation laws [2], based on the

concept of no-flow curves [1]. The new scheme is genuinely multidimensional in the sense that is Riemann solver

free which avoid dimensional splitting strategies. The full rigorous numerical analysis is carried out in [2]. In the

general context of multidimensional hyperbolic systems of conservation laws, the scheme satisfies the positivity

principle in the sense of the paper [3]. We also provide robust numerical examples to verify the theory and

illustrate the scientific computing capabilities of the proposed approach in advanced modeling and simulation.

1 Motivation and the Lagrangian-Eulerian formulation

To proceed with the construction of the semi-discrete Lagrangian-Eulerian scheme, consider the scalar 1D problem

∂u

∂t+∂H(u)

∂x= 0, x ∈ R, t > 0, u(x, 0) = u0(x), u0(x) ∈ L∞(R) where H ∈ C2(Ω), H : Ω → R, (1)

and u = u(x, t) : R× R+ −→ Ω ⊂ R. Following [1, 2], we obtain the fully discrete Lagrangian–Eulerian scheme,

un+1j = unj − ∆t

∆x

[F(unj , u

nj+1

)− F

(unj−1, u

nj

)], with a numerical flux function given by F

(unj , u

nj+1

)= (2)

1

4

[∆x

∆t

(unj −unj+1

)+∆x

fnj +fnj+1

∆xj

(unj +unj+1

)+

∆x2

4

fnj + fnj+1

∆xj

((ux)nj −(ux)nj+1

)+

∆x2

4∆t

((ux)nj + (ux)nj+1

)]. (3)

Thanks to the no-flow property

[∆x

∆t

]∝ [O(H(u)/u)], (see [1] and u and H(u) given by (1)), we can remove the

blow-up singularity of the numerical flux F (unj , unj+1) in (2)–(3) by replacing

∆x

∆twith a stability condition that

depends on O((H(u)/u)), which allows us to have ∆t → 0+ and produce an accurate approximation of the local

speeds. We set∆x

∆t∝ [O((H(u)/u))] in (2)–(3) for a suitable function

bj+ 12

= bj+ 12

(fj , fj+1), fj ≡H(uj)

uj≈ H(u)

ufor each j ∈ Z per time step [tn, tn+1]. (4)

Thus, the new class of SDLE schemes for hyperbolic-transport initial value problems (1) is given by

d

dtuj(t) = − 1

∆x[F (uj , uj+1) −F (uj−1, uj)], F (uj , uj+1)=

1

4

[bj+ 1

2

(u−j+ 1

2

−u+j+ 1

2

)+(fj + fj+1)

(u−j+ 1

2

+u+j+ 1

2

)],

where lim∆t−→0

F(unj , u

nj+1

)= ∞, with u−

j+ 12

= uj +∆x

4((ux)j) and u+

j+ 12

= uj+1 −∆x

4((ux)j+1). The formal

2D extension of the semi-discrete scheme is straightforward and given by

d

dtuj,k(t) = −

Fj+1/2,k −Fj−1/2,k

∆x−

Gj,k+1/2 − Gj,k−1/2

∆y, for the following scalar conservation law (5)

175

176

ut +H(u)x +G(u)y = 0, u(x, y, 0) = u0(x, y), where H,G,∈ C2, u0(x, y) ∈ L∞loc(R2). (6)

The corresponding multidimensional numerical fluxes in the x− and y−directions are, respectively, given by

Fj+ 12 ,k

=1

4

[bxj+ 1

2 ,k

(u−j+ 1

2 ,k− u+

j+ 12 ,k

)+ (fj,k + fj+1,k)

(u−j+ 1

2 ,k+ u+

j+ 12 ,k

)]and

Gj,k+ 12

=1

4

[byj,k+ 1

2

(u−j,k+ 1

2

− u+j,k+ 1

2

)+ (gj,k + gj,k+1)

(u−j,k+ 1

2

+ u+j,k+ 1

2

)], (7)

where the discretized multi-D (2D) space-time no-flow curves [1], given by (u, H(u), G(u) as defined in (6))

fj,k =H(ujk)

ujkand gj,k =

G(ujk)

ujk, with

[∆x

∆t

]∝ [O(H(u)/u)] and

[∆y

∆t

]∝ [O(G(u)/u)]. (8)

The intermediate values are given by u+j+1/2,k := uj+1,k(t) − ∆x4 (ux)j+1,k(t), u−j+1/2,k := uj,k(t) + ∆x

4 (ux)j,k(t),

u+j,k+1/2 := uj,k+1(t)− ∆y4 (uy)j,k+1(t), u−j,k+1/2 := uj,k(t) + ∆y

4 (uy)j,k(t), where numerical derivatives (ux)j,k(t) and

(uy)j,k(t) were computed via slope limiter approximations, and subject to the new no-flow CFL stability condition

maxj,k

(∆t

∆xmaxj,k

|fj,k|,∆t

∆ymaxj,k

|gj,k|)

≤ 1

4, without the need to employ the eigenvalues. (9)

Therefore, the extension for systems is straightforward (see [2]), and we will apply this version for systems using

the SDLE scheme (5)–(7) to numerically solve a 2D Euler system given by (e.g., [3]),ρt + (ρu)x + (ρv)y = 0,

(ρu)t + (ρu2 + p)x + (ρuv)y = 0,

(ρv)t + (ρuv)x + (ρv2 + p)y = 0,

Et + (u(E + p))x + (v(E + p))y = 0,(10)

where ρ is the mass density; u = u(x, y, t) and v = v(x, y, t), the x- and y-components of the velocity, respectively

and E = pe + 12ρ(u2 + v2). For a perfect gas, p = ρe(γ − 1), where constant γ denotes the ratio of specific heats;

and e, the internal energy of the gas. In all tests, we consider γ = 1.4 with the pre-and-post shock initial condition

(left, Double Mach Reflection) and the initial condition throughout the channel (right, A Mach 3 wind tunnel with

a step), where xs(t) = 10t/ sin(π/3) + 1/6 + y/ tan(π/3) is the shock position for the initial data (ρ, p, u, v)T0 =(1.4, 1, 0, 0)T , x > xs(0),

(8, 116.5, 4.125√

3,−4.125)T , x ≤ xs(0),or

(1.4, 1, 3, 0)T , x ≤ 0.6 and y ≥ 0,

(0, 0, 0, 0)T , x > 0.6, y > 0 and y ≤ 0.2,

(1.4, 1, 3, 0)T , x > 0.6 and y > 0.2,

(11)

References

[1] abreu, e. and perez, j., A fast, robust, and simple Lagrangian-Eulerian solver for balance laws and applications.

Computers and Mathematics with Applications, 77(9) (2019) 2310-2336.[2] abreu, e. and francois, j. and lambert, w. and perez, j., A class of positive semi-discrete Lagrangian-Eulerian

schemes for multidimensional systems of hyperbolic conservation laws, under review (in preparation for the second round

of peer review process).[3] lax, p. and Liu, x.-d. - Positivie schemes for solving multi-dimensional hyperbolic systems of conservation laws II,

Journal of Computational Physics, 187 (2003) 428-440.


A CONVERGENT FINITE DIFFERENCE METHOD FOR A TYPE OF NONLINEAR

FRACTIONAL ADVECTION-DIFFUSION EQUATION

JOCEMAR DE Q. CHAGAS1, GIULIANO G. LA GUARDIA2 & ERVIN K. LENZI3

1Departamento de Matematica e Estatıstica, UEPG, PR, Brasil, [email protected],2 Departamento de Matematica e Estatıstica, UEPG, PR, Brasil, [email protected],

3Departamento de Fısica, UEPG, PR, Brasil, [email protected]

Abstract

In this work, we apply a numerical method based on finite differences to solve a type of nonlinear advection-

diffusion fractional differential equation. The fractional operator considered is the fractional Riemann-Liouville

derivative or the fractional Riesz derivative of order α, with 1 < α ≤ 2. The nonlinearity is of type uν(x, t), with

ν > 0, on which the spatial fractional derivative acts.

1 Introduction

In this work, we apply a finite difference method to solve a nonlinear advection-diffusion fractional equation of the

form

∂

∂tu(x, t) = −a(x)

∂

∂xu(x, t) + b(x)

∂α

∂|x|αuν(x, t) , (1)

where t > 0, x ∈ I = [L,R] ⊆ R, and a(x), b(x) ≥ 0 for all x ∈ I. Only positive exponents ν are considered. The

fractional operator ∂α

∂|x|α is the fractional Riemann-Liouville derivative or the fractional Riesz derivative of order

α (for details, see Refs. [1, 2, 3]), where 1 < α ≤ 2. For simplicity, in the main result, we only consider the case

a(x) = b(x) = 1 for all x.

The method used was introduced in [4], proposed for solving the nonlinear fractional diffusion equation

∂

∂tu(x, t) = c2

∂α

∂|x|αuν(x, t), (2)

where the function u(x, t) ≥ 0 is unknown, c2 represents the diffusion coefficient, and the fractional operator is

considered with 1 < α ≤ 2. The exponent ν = 1 is usually related to unusual relaxation processes and allows

us to deal with problems related to anomalous diffusion processes (see, for instance, Ref [1]). The implicit Euler

method developed holds for ν > 0 and it is consistent and unconditionally stable, therefore convergent by Rosinger’s

Theorem [5], which is a nonlinear extension of the celebrated Lax-Richtmyer equivalence theorem. Such a method

was also applied in a recent work [6].

2 Main Result

The numerical approach is the standard for finite differences. We consider a mesh with 0 ≤ j ≤M and 0 ≤ n ≤ N .

The exact solution u(x, t) evaluated in the grid point (xj , tn) is denoted by unj . The boundary values of the domain

are x0 = L and xM = R, and tN = t denotes the final time. To denote the numerical solutions, we utilize the

notation v(xj , tn) or vnj .

Assuming that ν > 0 and δ > −1, such that ν = 1 + δ, we multiply Eq. (1) by (1 + δ)uδ(x, t), yielding

∂

∂tuν(x, t) = −a(x)

∂

∂xuν(x, t) + ν uδ(x, t) b(x)

∂α

∂|x|αuν(x, t) . (3)

177

178

In the grid points, we denote the weights ν vδ(xj , tn) only by δnj := ν (vδ)

n

j = ν vδ(xj , tn). Our main result is

the implicit Euler method (4), given in the next Theorem.

Theorem 2.1. The implicit Euler method

−λδn+1j wα

0 (vν)n+1j+1 +(1−λδn+1

j wα1 + λ1)(vν)

n+1j − (λδn+1

j wα2 + λ1)(vν)

n+1j−1−λδ

n+1j

j+1∑i=3

wαi (vν)

n+1(j+1)−i = (vν)

nj (4)

where λ = khα , λ1 = k

h and δnj = ν (vδ)n

j , to solve the Eq. (3) with 1 < α ≤ 2, ν > 0 and a(x) = b(x) = 1, on

the finite domain L ≤ x ≤ R, together with a non-negative bounded initial condition u(x, 0) = u0 and boundary

conditions u(x = L, t) = 0 = u(x = R, t) for all t ≥ 0, based on the shifted GrA¼nwald approximation (see it in [2]

or in [3]), with p = 1 and h = (R− L)/M , is consistent and unconditionally stable.

Proof It is sufficient to adapt the proof of Theorem 3.1 of [4], an extension for the nonlinear case of Theorem 2.7

of Ref. [7].

Corollary 2.1. The implicit Euler method (4) is convergent.

Proof Just apply the Theorem presented by Rosinger in Ref. [5].

We observe that one cannot apply directly the implicit Euler method (4) since the weights δn+1j are considered

in the (n + 1)-step. To apply the method shown in Eq. (4), it is necessary to compute the value of the weights

δn+1j in each time-step. We propose an iteration procedure in which each step is evaluated twice: using the known

weights to one time-step ago, we evaluate the weights δj to the next time-step, and then we return and evaluate

the unknown function at the next time-step.

References

[1] evangelista, l.r.; lenzi, e.k. - Fractional Diffusion Equations and Anomalous Diffusion. Cambridge

University Press, 2018.

[2] karniadakis, g.e. (ed) - Handbook of Fractional Calculus with Applications, vol. 3: Numerical Methods. De

Gruyter, 2019.

[3] li, c.; chen, a. - Numerical methods for fractional partial differential equations. International Journal of

Computer Mathematics, 95, 6-7, 1048-1099, 2018.

[4] chagas, j.q.; la guardia, g.g.; lenzi, e.k. - A finite difference method for a class of nonlinear fractional

advection-diffusion equations. Partial Differential Equations in Applied Mathematics, 4, 100090, 2021.

[5] rosinger, e.e. - Stability and Convergence for Non-Linear Difference Scheme are Equivalent. Journal of the

Institute of Mathematics and its Applications., 26, n. 2, 143-149, 1980.

[6] la guardia, g.g.; chagas, j.q.; lenzi, m.k.; lenzi e.k. - Solutions for Nonlinear Fractional Diffusion

Equations with Reactions Terms. In: Singh, H.; Singh, J.; Purohit, S. D.; Kumar, D. - Advanced Numerical

Methods for Differential Equations, CRC Press, 2021.

[7] meerschaerdt, m.m.; tadjeran, c. - Finite difference approximations for fractional advection-dispersion

flow equations. Journal of Computational and Applied Mathematics, 172, 65-77, 2004.


SOLUTION OF LINEAR RADIATIVE TRANSFER EQUATION IN HOLLOW SPHERE BY

DIAMOND DIFFERENCE DISCRETE ORDINATES AND ADOMIAN METHODS

MARCELO SCHRAMM1, CIBELE A. LADEIA2 & JULIO C. L. FERNANDES3

1PPGMMat, UFPel, RS, Brasil, [email protected],2 PPGMAp, UFRGS, RS, Brasil, [email protected],3PPGMAp, UFRGS, RS, Brasil, [email protected]

Abstract

In this work, a methodology to solve radiative transfer problems in spherical geometry without other forms of

heat exchange is presented. The authors used a decomposition method, based on the Adomian routines, together

with a diamond difference scheme. The algorithm is simple, highly reproducible and can be easily adapted to

further problems or geometries. Also, the authors introduce a brief necessary criterion for convergence and

consistency using an algebraic residual term analysis. The numerical results are compared with some classical

and recent cases in the literature, along with a simplified version of a complete (fully coupled with heat exchange

problem) case.

1 Introduction

In this work we present a hybrid methodology with application to a test case with a focus on formalism and a

convergence criterion. One of the main objectives here is to take an initial step in this sense and present results

that analyze and guarantee the convergence of the method by the quick decay of the algebraic residuals.

We consider the radiative transfer equation in spherical geometry for hollow sphere [1],

µ

r2∂

∂r

[r2I(r, µ)

]+

1

r

∂

∂µ

[ (1 − µ2

)I(r, µ)

]+ I(r, µ) =

(1 − ω (r)

)Ib (T ) +

ω (r)

2

∫ 1

−1

p (µ, µ′) I (r, µ′) dµ′ , (1a)

where r ∈ [R1, R2] is the optical space variable and µ ∈ [−1, 1] is the direction cosine. R1 and R2 are the radii of

the inner and outer spherical surfaces, respectively. Further, I(r, µ) is the radiation intensity, Ib (T ) is the black

body radiation for temperature T , ω is the single scattering albedo and p (µ, µ′) is the phase function [2]. The

boundary conditions of Equation (1a) are

I(Rk, (−1)

k+1µ)

= ϵkIbk(T ) + ρk

∫ 1

0

I(Rk, (−1)

kµ′)µ′dµ′ , 0 < µ ≤ 1 (1b)

for k ∈ 1, 2, where ϵ1 and ϵ2 are the emissivities of the inner and outer surfaces, respectively. In the same way, ρ1

and ρ2 are the diffusive reflectivities for the inner and outer surfaces, respectively. Ib1(T ) and Ib2(T ) are the black

body radiations for inner and outer surfaces in temperature T , respectively.

To solve (1) we implement the discrete ordinates method, evaluating the equations in certain µ = µm. The

derivative with respect to µ is approximated using a diamond difference scheme and the integral is evaluated using

the Gauss-Legendre quadrature rule. The abscissas of this quadrature rule are the discrete ordinates µm, and we

define I (r, µm) = Im (r). After, we used the decomposition method, which briefly consists in expanding Im as an

infinite series. For computational purposes and due to the necessity of the application of a numerical integration

scheme to solve the recursive equations, we segment the domain in N + 1 nodes rı, define Iım = Im (rı) and the

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180

decomposed solution writes

Iım =

n∑ȷ=0

(Iım

)ȷ. (2)

We made a recursive system among the (Iım)ȷ using (2) in (1) like

AIȷ = BIȷ−1 (3)

for ȷ = 1, 2, . . . , n. By their solution and (2) we reconstruct Iım. Here, Iȷ are (Iım)ȷ in vector notation, A and B

are constant two-dimensional arrays. Also, in (1) we considered the terms (1 − ω (r)) Ib (T ) and ϵkIbk (T ) for ȷ = 0

only.

We demonstrate consistency of this numerical scheme by setting an upper bound to the residual term in (1),

substituting (2), using some norm operations and proving it goes to zero as n increases. In addition, we present a

necessary condition to the convergence of (2) using the divergence test.

2 Main Results

Several cases of [3] were successfully solved using the presented methodology, with small differences in the numerical

results. Using (3) with ȷ→ ∞ would result in an infinite series in (2), yielding an exact representation of Iım, however

using a truncated series, a remaining (residual) term remains from the substitution. Substituting (2) in (1), taking

the maximum norm and using the triangle inequality, we obtain

∥εn∥∞ ≤ ∥C∥∞ ∥In∥∞ . (4)

where εn is the vector notation for the residual term using the truncated series (2), and C is a constant two-

dimensional array. As C does not vary with n, ∥εn∥∞ is majored by a constant scale of ∥In∥∞. In other words,

if ∥In∥∞ → 0, then ∥εn∥∞ → 0 and the method is consistent. Now, using norm operations and (3), we see that

∥In∥∞ → 0 as ȷ = n→ ∞ if

∥A∥∞ > ∥B∥∞ . (5)

This inequality is also a necessary condition for convergence of the series in (2).

The combination of the Adomian decomposition method with the discrete ordinates method yield results that

did not differ from the reference by a significant amount, and for the test cases, the processing time did not

exceed five seconds in a domestic computer. Despite its simplicity, this research sets the basis for the complex and

time demanding research towards the convergence of the Adomian decomposition method, absent in the literature.

We are still developing closed and sufficient criteria for other problems involving the application of the Adomian

decomposition method in transport problems, and we intend to do it for some usual non-linear terms, like the

coupling with the heat diffusion equation.

References

[1] Ladeia, C. A. and Schramm, M. and Fernandes, J. C. L. - A simple numerical scheme to linear radiative

transfer in hollow and solid sphere. Semin., CiAªnc. Exatas Tecnol., 41, 21-30, 2020.

[2] Chandrasekhar, S. - Radiative Transfer., Oxford University Press, New York, 1950.

[3] Abulwafa, E. M. - Radiative-transfer in a linearly-anisotropic spherical medium. J. Quant. Spectrosc. Radiat.

Transfer, 49, 165-175, 1993.


ANALYSIS RESULTS ON AN ARBITRARY-ORDER SIR MODEL CONSTRUCTED WITH

MITTAG-LEFFLER DISTRIBUTION

NOEMI ZERAICK MONTEIRO1 & SANDRO RODRIGUES MAZORCHE2

1Departamento de Matematica, UFJF, MG, Brasil, [email protected],2Departamento de Matematica, UFJF, MG, Brasil, [email protected]

Abstract

Our recent works discuss the construction of a meaningful arbitrary-order SIR model. We believe that

arbitrary-order derivatives may arise from potential laws in the infectivity and removal functions. This work

intends to summarize previous results, as well as show new results on a model with Mittag-Leffler distribution.

We emphasise our optimization process, the nonlocality of the model and the behavior near the lower terminal.

1 Introduction

Arbitrary-Order Calculus, commonly known as Fractional Calculus, is a great tool for describe the dynamic of

many processes, mainly because of its “memory effect”. Generally, the models are obtained by replacing a integer

derivative with an arbitrary-order one. Compartmental models, for example, have been widely studied with arbitrary

orders. We investigate the use of arbitrary orders in SIR-type models, theoretically, analytically and numerically.

Recalling that a model is constructed by modelling the physical process, we ask what features are maintained when

exchanging the orders. Are consistent models established, regarding the definition of parameters, physical meaning

etc.? We need to give attention to how, where, and why the arbitrary-orders interfere in the model.

2 Main Results

Arbitrary-Order Calculus, commonly known as Fractional Calculus, is a great tool for describe the dynamic of

many processes, mainly because of its “memory effec”. Generally, the models are obtained by replacing an integer

derivative with an arbitrary-order one. Compartmental models, for example, have been widely studied with arbitrary

orders. We investigate the use of arbitrary orders in SIR-type models, theoretically, analytically and numerically.

Recalling that a model is constructed by modelling the physical process, we ask what features are maintained when

exchanging the orders. Are consistent models established, regarding the definition of parameters, physical meaning

etc.? We need to give attention to how, where, and why the arbitrary orders interfere in the model.

3 Main Results

As discussed in [1], so far we have not been able to find a physical-based modelling that simply allows to

change the orders of the derivatives. However, arbitrary orders can be obtained through potential laws in the

infectivity and removal functions. We present in [2] a physical derivation of an arbitrary-order model, following

[3], with the language of the Continuous Time Random Walks (CTRW). The individual’s removal time from the

infectious compartment follows a Mittag-Leffler distribution related to α, while the parameter β is related to the

infectivity function. The Riemann-Liouville derivative arises from the modelling and the arbitrary-order model

with 1 ≥ β ≥ α > 0 is given by (1)-(3), where γ(t) is the vital dynamic; ω(t), the extrinsic infectivity; N , the total

population; τ , a scale parameter and, θ(t, t′), the probability that an infectious since t′ has not died of natural death

until t. If β = α = 1 and γ(t), ω(t) are taken constants, we get the classic SIR model. Note that dN(t)/dt = 0, so

181

182

the population is constant. In [2], we revisited the work and used optimization to apply it to COVID-19 pandemic

data.dS(t)

dt= γ(t)N − ω(t)S(t)θ(t, 0)

NτβD1−β

(I(t)

θ(t, 0)

)− γ(t)S(t), (1)

dI(t)

dt=ω(t)S(t)θ(t, 0)

NτβD1−β

(I(t)

θ(t, 0)

)− θ(t, 0)

ταD1−α

(I(t)

θ(t, 0)

)− γ(t)I(t), (2)

dR(t)

dt=θ(t, 0)

ταD1−α

(I(t)

θ(t, 0)

)− γ(t)R(t), (3)

We set up a L1-scheme based discretization to implementation on MATLAB and, for optimization, a Feasible

Direction Interior Point Algorithm. In [4], we are doing parameter analysis in the model (1)-(3). Also, in [5], we

are dealing with equilibrium, reproduction numbers, monotonicity and non-negativity, while in [6] we deal with

pandemic data in Brazilian states. Here, we pretend to summarize the equilibrium characterization, present some

considerations on the use of FDIPA and also deal with two points that were not discussed: the model is nonlocal

and presents a nonintuitive behavior in the lower terminal. In fact, given the asymptotic behavior of the derivatives

near the lower terminal, if β > α, one have dI/dt < 0 for t sufficiently small, as illustrated in Figure 1. About

the nonlocality, in the classic SIR model, epidemiological parameters define the epidemic independently of time:

for each point, there is a unique trajectory. However, this is not valid for arbitrary-order models. The formulation

of the IVP disregards the past, but, once the model is nonlocal, this modifies the trajectory. We illustrate this

considering N = 1000000, initial conditions S(0) = N − 1, I(0) = 1 and R(0) = 0 and dt = 0.1. At time t = 90,

we consider the initial condition given by S(90), I(90), R(90) and run the model again. In Figure 2, we have the

equivalent trajectories for a maximum time T = 3000. The equilibrium is the same, but the trajectories are not.

0 0.2 0.4 0.6 0.8 1

t in days

0.95

1

1.05

1.1

1.15

I

Parameters: !=5; ==15; .=0.01,=0.5; -=0.9

Figure 1: Behaviour Lower Terminal.

0 2 4 6 8 10

S #10 5

0

0.5

1

1.5

2

2.5

3

I

#10 5Parameters: !=3; ==10; .=0.01

,=0.7; -=0.9

Figure 2: Change on trajectory.

So, one should to start the simulation of an epidemic on its beginning or be able to say about the past, what

is a trick question. We aim that the deeper study of the equilibrium points and trajectories are fundamental to

predict important features of the epidemic. By other hand, the mathematics of the Fractional Calculus’ models is

still a black box of surprises that the assembling between analytic and numerical studies can help to investigate.

References

[1] s. r. mazorche and n. z. monteiro - Modelos epidemiologicos fracionarios: o que se perde, o que se ganha,

o que se transforma?. Proceeding Series of the XL CNMAC, 2021. (to appear)

[2] n. z. monteiro and s. r. mazorche - Fractional derivatives applied to epidemiology. Trends in

Computational and Applied Mathematics, vol. 0, n. 0, pp. 157-177, 2021.

[3] c. n. angstmann, b. i. henry, and a. v. mcgann - A fractional-order infectivity and recovery sir model.

Fractal and Fractional, vol. 1, n. 1, pp. 11, 2017.

[4] n. z. monteiro and s. r. mazorche - Estudo de um modelo SIR fracionario construıdo com distribuicao de

Mittag-Leffler. Poster presentation. Brazilian Society of the XL CNMAC, 2021. (to appear)

[5] n. z. monteiro and s. r. mazorche - Analysis and application of a fractional SIR model constructed with

Mittag-Leffler distribution. Proceeding Series of the XLII CILAMCE, 2021. (to appear)

[6] n. z. monteiro and s. r. mazorche - Application of a fractional SIR model built with Mittag-Leffler

distribution. Poster presentation. Mathematical Congress of the Americas, 2021.


NUMERICAL ANALYSIS FOR A THERMOELASTIC DIFFUSION PROBLEM IN MOVING

BOUNDARY

RODRIGO L. R. MADUREIRA1 & MAURO A. RINCON2

1NCE, UFRJ, RJ, Brasil, [email protected],2Instituto de Matematica, UFRJ, RJ, Brasil, [email protected]

Abstract

In this work, error estimates are shown for the semi-discrete and totally discrete problems of a linear

thermoelastic diffusion model with a moving boundary, considering the null boundary condition. The resulting

linear system is solved through three numerical methods: Coupled, Uncoupled with Predictor-Corrector and

Uncoupled. The order of convergence obtained in the L∞(0, T ;L2(Ω)) and L∞(0, T ;H10 (Ω)) norms is consistent

with the theoretical results.

1 Introduction

Consider the following thermoelastic diffusion problem:∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

ρ∂2u

∂t2− α

∂2u

∂x2+ γ1

∂θ

∂x+ γ2

∂P

∂x= f1(x, t), in Q,

c∂θ

∂t+ d

∂P

∂t− k

∂2θ

∂x2+ γ1

∂2u

∂x∂t= f2(x, t), in Q,

η∂P

∂t+ d

∂θ

∂t− h

∂2P

∂x2+ γ2

∂2u

∂x∂t= f3(x, t), in Q,

u = θ = P = 0, in Σ,

u(x, 0) = u0(x), u′(x, 0) = u1(x), θ(x, 0) = θ0(x), P (x, 0) = P0(x), ∀x ∈ [−K(0),K(0)],

(1)

where the domain Q ⊂ R2 is defined by

Q = (x, t) ∈ R× (0, T );x ∈ It and It = x ∈ R;−K(t) < x < K(t),

and its lateral boundary is given by Σ =⋃

0<t<T

(−K(t),K(t)) × t, where K(t) is a function K : [0, T ] → R,

which defines the moving boundary. The solution u, θ, P of the problem (1) is composed of functions which

depend on the spatial and temporal variables x and t, respectively, that is, u = u(x, t), θ = θ(x, t), P = P (x, t).

The apostrophe symbol, ′, denotes the partial derivative with respect to t.

We will show the theoretical and numerical aspects of the (1) problem in order to establish the error estimate

of solutions in Sobolev spaces for the discrete problem and semi-discrete problem, as well as perform numerical

simulations to analyze the behavior of the solution, order of numerical convergence and errors. For theoretical

development, the Faedo-Galerkin method and interpolation theory results are applied to obtain an inequality from

the approximate solution to the exact solution. To obtain the numerical solution, the finite element method is

applied to the spatial variable and the finite differences method to the temporal variable. The system of ordinary

differential equations resulting from the application of these methods is naturally coupled. For the numerical

solution of this system, we developed the Coupled, Uncoupled with Predictor-Corrector and Uncoupled methods.

183

184

2 Main Results

2.1 Assumptions

The function K(t) and the constants satisfy the following assumptions,

(H1) K ∈ C3([0, T ];R), with K0 = min0≤t≤T

K(t) > 0.

(H2) |K ′′(t)| ≤ cK(t), ∀t ∈ [0, T ], where c > 0.

(H3) There is a positive constant K1 < 1, such that |K ′(t)| ≤ K1.

(H4) ρ, α, γ1, γ2, k and h are strictly positive.

(H5) The constants c, d and η are positive and satisfy cη − d2 > 0.

Condition (H5) is necessary for stability of the problem (1). The existence and uniqueness of the solution

u, θ, P of the problem (1) are known in the literature (see [1]).

2.2 Discrete error estimates in the H10 (Ω) and L2(Ω) norms

Under the conditions (H1)-(H5) and conveniently chosen initial data, we show that the error estimate in the H10 (Ω)

norm has convergence order O(h + ∆t2) and the error estimate in the L2(Ω) norm has an order of convergence

O(h2 + ∆t2) (see [2]). Here, Ω = (−1, 1). Furthermore, h and ∆t are, respectively, the mesh sizes of space and

time.

2.3 Numerical simulations

Numerical simulations are shown using three numerical methods developed for the resulting system of ordinary

differential equations: Coupled, Uncoupled with Predictor-Corrector, and Uncoupled. They all prove the theoretical

convergence order, however with very different execution times (see [3, 3]).

References

[1] aouadi, m. and soufyane, a. - Polynomial and exponential stability for one-dimensional problem in

thermoelastic diffusion theory. Appl. Anal., 89(6), 935-948, 2010.

[2] madureira, r., rincon, m.a. and aouadi, m. - Numerical analysis for a thermoelastic diffusion problem in

moving boundary. Mathematics and Computers in Simulation, 187, 630-655, 2021.

[3] madureira, r.l.r., rincon, m.a. and aouadi, m. - Global existence and numerical simulations for a

thermoelastic diffusion problem in moving boundary. Mathematics and Computers in Simulation, 166, 410-

431, 2019.

[4] madureira, r.l.r., rincon, m.a. and teixeira, m.g. - Numerical methods for a problem of thermal

diffusion in elastic body with moving boundary. Numerical Methods for Partial Differential Equations, 37(4),

2849-2870, 2021.


NUMERICAL ANALYSIS AND TRAVELLING WAVE SOLUTIONS FOR AN INTERNAL WAVE

SYSTEM

WILLIAN C. LESINHOVSKI1 & AILIN RUIZ DE ZARATE2

1Programa de Pos-Graduacao em Matematica, UFPR, PR, Brasil, [email protected],2Departamento de Matematica, UFPR, PR, Brasil, [email protected]

Abstract

In this work we focus on approximations of travelling wave solutions for a nonlinear system of Boussinesq type

with a nonlocal operator. For numerical purposes, we focus on the case where the solutions are periodic functions

in space with period 2l > 0. Three approaches to calculate travelling waves are proposed and compared. For

this an efficient and stable scheme for the nonlinear system, based on a von Neumann stability analysis for the

linearized problem, is used to capture the evolution of approximate travelling wave solutions. Also, a scheme for

the corrugated bottom version of the nonlinear system is proposed and validated.

1 Introduction

Asymptotic analysis of the Euler equations is a successful method for the study of internal ocean waves. For the case

of intermediate depth for the lower layer and shallow upper layer, a strongly nonlinear model for internal waves was

obtained in [1]. It describes the evolution of the interface η(x, t) between the fluids and the upper layer averaged

horizontal velocity u(x, t), where x and t represent the spatial and temporal variables, respectively. Considering a

weakly nonlinear wave propagation regime, flat bottom and in nondimensional variables that system reads:ηt −

[(1 − αη)u

]x

= 0,

ut + αuux − ηx =√βρ2ρ1

Tδ [u]xt +β

3uxxt.

(1)

The pseudo-differential operator Tδ is the Hilbert transform on the strip and α, β, ρ1 and ρ2 are positive

constants where α = O(β).

2 Main Results

For the discretization of the nonlinear system we consider its linearization around the zero equilibrium to implement

the method of lines. A fourth order finite difference scheme for spatial derivatives and a spectral approach for the

dispersive terms are considered in the semi-discretization and the classical fourth order Runge-Kutta (RK4) scheme

is used for time advancing. The stability conditions obtained in a von Neumann analysis are validated in numerical

tests and extended to the scheme for the nonlinear system (1) which includes the discretization of the nonlinear

terms α(ηu)x and αuux as presented in [2].

Initially, we consider as initial condition for the nonlinear system the traveling wave solutions of the Intermediate

Long Wave (ILW) equation and its regularized version (rILW). Both waves perform satisfactorily preserving their

shapes in a given time interval as presented in [4]. In addition, three approaches to calculate travelling waves for

the nonlinear system (1) are proposed in [3]. Supposing that the system admits a travelling wave solution, we define

the variable y = x− ct, integrate both equations on y and consider the integration constant to be equal to zero to

obtain

185

186

− cη − (1 − αη)u = 0,

− cu+α

2u2 − η + c

√βρ2ρ1

Tδ[uy]

+ cβ

3uyy = 0.

(2)

For the first approach we reduce system (2) to an equation on η using a second order Taylor approximation.

For the second approach we obtain an equation on u from system (2) using algebraic computations. For the third

approach we take the complete system (2). In all approaches we consider the wave speed c as an unknown variable

and complete the problem with the conservation law∫ l

−l

η(y) dy = d,

from system (1), where d is a constant.

The discretization is done considering an uniform grid on the interval [0, 2l] and the resulting system of equations

is solved by the Newton’s method using the traveling wave of the rILW equation as initial guess. The first approach

did not improve the results of the initial guess and the second approach presented profiles that perform worse than

the initial ones. On the other hand the third approach improved the results obtained in [4] and proved to be a good

method to obtain travelling waves for system (1).

In the last part of the work we consider a more general case of the intermediate wave model (1), where there is

an irregular topography on the bottom that can be described by a variable coefficient in a nonlinear system given

in the computational domain (ξ, t) byηt =

1

M(ξ)

[(1 − αη)u

]ξ,

ut +α

M(ξ)uuξ −

1

M(ξ)ηξ =

ρ2ρ1

√β

M(ξ)T[u]ξt

+β

3M(ξ)

(uξtM(ξ)

)ξ

.

(3)

This formulation allowed us to propose a numerical method for system (3) based on the one for the nonlinear

system (1). The effects of the topography in the solutions and in the stability conditions are illustrated and

compared with the solutions for the flat bottom cases.

References

[1] ruiz de zarate, a. and alfaro vigo, d. and nachbin, a. and choi, a. - A Higher-order Internal Wave

Model Accounting for Large Bathymetric Variations. Studies in Applied Mathematics, 122, 275-294, 2009.

[2] lesinhovski, w. c. - Analise de von Neumann de um modelo dispersivo de ondas internas, Dissertacao de

Mestrado, UFPR, 2017.

[3] lesinhovski, w. c. - Numerical analysis and approximate travelling wave solutions for an internal wave

system, Tese de Doutorado, UFPR, 2021.

[4] lesinhovski, w. c. and ruiz de zarate, a. - Numerical analysis and approximate travelling wave solutions

for a higher order internal wave system. Accepted for publication in Trends in Computational and Applied

Mathematics, 2021.


IMPROVED REGULARITY FOR NONLOCAL ELLIPTIC EQUATIONS THROUGH ASYMPTOTIC

PROFILES

AELSON O. SOBRAL1 & DISSON DOS PRAZERES2

1Federal University of Paraıba, UFPB, PB, Brazil, [email protected],2Federal University of Sergipe, UFS, SE, Brazil, [email protected]

Abstract

We obtain improved regularity for viscosity solutions of a special class of nonlocal L0(σ)-elliptic equations.

More precisely, we introduce the notion of recession operator to the nonlocal setting, discuss its main features

and then apply a compactness method to transfer regularity from asymptotic profiles. The role of this class is

confined into some examples and a C1,(σ−1)− regularity estimate.

1 Introduction

In this work, we are concerned with improved regularity estimates for viscosity solutions w of

I[w] = f(x) ∈ L∞(B1), (1)

where I is a L0(σ)-elliptic operator under an asymptotic regime. Here, L0(σ)-ellipticity means that the inequality

M−L0(σ)

[u− v](x) ≤ I[u](x) − I[v](x) ≤ M+L0(σ)

[u− v](x),

holds whenever they are well-defined, where

M−L0(σ)

[u](x) = infL∈L0(σ)

L[u](x), M+L0(σ)

[u](x) = supL∈L0(σ)

L[u](x),

and L0(σ) is the class of linear operators of the form

L[u](x) =

∫Rn

(u(x+ y) + u(x− y) − 2u(x))K(y)dy,

where K is symmetric and satisfies λ(2 − σ) ≤ |y|n+σK(y) ≤ Λ(2 − σ).

The background for the search of qualitative properties of solutions to nonlocal equations of the type (1) have

gained much attention since the seminal work [1] of Caffarelli and Silvestre in 2009. So far, a lot of interesting

studies have emerged, for example, [2], from the same authors, in which they establish compactness results for

equations like (1) and extend the results from [1] to equations with x-dependence. They use a scalling argument

that relies on the invariance of the class of ellipticity. Therefore, to obtain Holder regularity for the gradient of the

solutions, they need to consider proximity to an operator on the subclass L1(σ) ⊂ L0(σ), such that ∇K decay like

the gradient of |y|−n−σ−1.

A few years later, in [3], Kriventsov establishes C1,β for some unknown β to equations like (1) with x-dependence

and merely bounded data using a pertubative argument. One can also find further regularity results when the main

operator is concave, or with better boundary regularity datum.

The key novelty of our paper is to bring the notion of recession operator to the nonlocal context(it was already

introduced in the local setting, see for example [3]), discuss it properly and adapt the strategy in [2] to transfer

regularity from the asymptotic profile of the main operator in (1). Under a special asymptotic regime, we are able to

improve regularity up to C1,(σ−1)− , which is almost optimal. Furthermore, we don’t need the presence of concavity

or better regularity on the boundary values.

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188

2 Main Results

Given a nonlocal operator I as in (1), we define the nonlocal recession operator I∗ as a weakly subsequential limit(in

the weak sense of [2]) of the family Iµµ>0 as µ→ 0, where Iµ = µI[µ−1−]. This can be done by stability results

from [2], which assures that the nonlocal recession operator I∗ always exists and is, at least, on the same class of

ellipticity of the original operator.

Under uniqueness assumptions of the convergence above, we are able to give some qualitative aspects to the

nonlocal recession operator such as

Homogeneity of degree 1;

Rate of convergence of the family Iµµ>0 as µ→ 0.

Besides that, through an adaptation of the compactness strategy written by Caffarelli and Silvestre in [2], we are

able to transfer regularity from the limiting profile to the solutions of the original equation, and so, the following

theorem comes out

Theorem 2.1. Let σ > 1, w ∈ C(B1) ∩ L∞(Rn) be a viscosity solution of (1) with I in the following asymptotic

regime:

(I) I is L0(σ)-elliptic and is translation invariant;

(II) The recession operator I∗ has Cσ estimates;

(III) For all µ > 0 and 0 < λ < 1 there holds

λσµI[µ−1v(λ−1−)](λ−) = λσµI[µ−1λ−σv](−).

Then w ∈ C1,α(B1/2) for every α ∈ (0, 1) such that 1 + α < σ and hold the estimate

||w||C1,α(B1/2) ≤ C(||w||L∞(Rn) + ||f ||L∞(B1)

),

where C depends on dimension, α, σ and the constant from the Cσ estimates of I∗.

References

[1] caffarelli, l. and silvestre, l. - Regularity theory for fully nonlinear integro-differential equations.,

Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical

Sciences, 62(5), 597-638, 2009.

[2] caffarelli, l. and silvestre, l. - Regularity results for nonlocal equations by approximation., Archive for

Rational Mechanics and Analysis, 200(1), pp.59-88, 2011.

[3] kriventsov, d. - C1,α interior regularity for nonlinear nonlocal elliptic equations with rough kernels.,

Communications in Partial Differential Equations, 38(12), 2081-2106, 2013.

[4] silvestre, l. and teixeira, E. Regularity estimates for fuly non linear elliptic equations that are

asymptotically convex., Prog. Nonlinear Differential Equations, 86, 425-438, 2019.


EXISTENCE AND NONEXISTENCE OF SOLUTION FOR A CLASS OF QUASILINEAR

SCHRODINGER EQUATIONS WITH CRITICAL GROWTH

DIOGO DE S. GERMANO1 & UBERLANDIO B. SEVEREO2

1UAMat, UFCG, PB, Brasil, [email protected],2DM, UFPB, PB, Brasil, [email protected]

Abstract

In this work, we study the existence and nonexistence of solution for the following class of quasilinear

Schrodinger equations:

−div(g2(u)∇u) + g(u)g′(u)|∇u|2 + V (x)u = f(x, u) + h(x)g(u) in RN ,

where N ≥ 3, g : R → R+ is a continuously differentiable function, V (x) is a potential that can change sign, the

function h(x) belongs to L2N/(N+2)(RN ) and the nonlinearity f(x, s) is possibly discontinuous and may exhibit

critical growth. In order to obtain the nonexistence result, we deduce a Pohozaev identity and the existence of

solution is proved by means of a fixed point theorem.

1 Introduction

We consider the following class of quasilinear elliptic equations:

− div(g2(u)∇u) + g(u)g′(u)|∇u|2 + V (x)u = f(x, u) + h(x)g(u) in RN , (1)

where N ≥ 3, g : R → R+ is a C1-class function, V : RN → R is a potential that can change sign, f : RN ×R → Ris a measurable function, which may have critical growth and h ∈ L2N/(N+2)(RN ), h = 0. This work is based on

the article [4].

The study of equation (P) is related with the existence of standing wave solutions for quasilinear Schrodinger

equations of the form

i∂tw = −∆w +W (x)w − p(x, |w|2)w − ∆[ρ(|w|2)]ρ′(|w|2)w, (2)

where w : R × RN → C is the unknown, W : RN → R is a given potential, ρ : R+ → R and p : RN × R+ → Rare real functions satisfying appropriate conditions. Equation (2) is called in the current literature as Generalized

Quasilinear Schrodinger Equation and it has been accepted as model in many physical phenomena depending on

the function ρ. If we take g2(u) = 1 + [(ρ(u2))′]2

2 , then (2) turns into quasilinear elliptic equation (P) (see [5]).

Furthermore, depending on the form of the function g, equation (P) can take several forms already well known in

the literature, such as

−∆u+ V (x)u = p(x, u) in RN ,

−∆u+ V (x)u− ∆(u2)u = p(x, u) in RN ,

−∆u+ V (x)u− γ∆(|u|2γ)|u|2γ−2u = p(x, u) in RN ,

or

−∆u+ V (x)u− ∆[(1 + u2)1/2]u

2(1 + u2)1/2= p(x, u) in RN .

189

190

Motivated by these physical and mathematical aspects, equation (P) has attracted a lot of attention of many

researchers and some existence and multiplicity results have been obtained, see [1, 2, 3, 5] and references therein.

In this work, we intend to prove a Pohozaev identity for the equation

− div(g2(u)∇u) + g(u)g′(u)|∇u|2 + V (x)u = p(u) in RN (3)

and, as a consequence of the identity, to exhibit the critical exponent for this type of equation. Moreover, under

convenient conditions on g(s), V (x), f(x, s), h(x) and by applying a fixed point theorem, we show that equation

(P) admits at least one weak solution.

2 Main Results

Theorem 2.1 (Pohozaev identity). Suppose that u ∈ C2(RN ) is a classical solution for problem (3), with g ∈ C1(R),

V ∈ C1(RN ,R) and p ∈ C(R). Moreover, assume that∫RN

[g2(u)|∇u|2 + (|x · ∇V (x)| + |V (x)|)u2 + |P (u)|

]dx <∞, (1)

where P (s) =∫ s

0p(τ)dτ . Then, u satisfies the identity

N − 2

2

∫RN

g2(u)|∇u|2dx+N

2

∫RN

V (x)u2dx+1

2

∫RN

[x · ∇V (x)]u2dx

= N

∫RN

P (u)dx.

(2)

Proof To show this Pohozaev identity see [4].

Theorem 2.2 (Result of Existence). Assume appropriate conditions on the functions g, V and f . Furthermore,

assuming that h ∈ L2N/(N+2)(RN ), there exists δ0 > 0 such that if ∥h∥2N/(N+2) ≤ δ0 then equation (P) has at least

a weak solution.

Proof To know the proper assumptions about g, V and f and to prove this result of existence, see [4].

References

[1] deng, y., peng, s. and yan, s. - Critical exponents and solitary wave solutions for generalized quasilinear

Schrodinger equations J. Differential Equations, 260, 1228-1262, 2016.

[2] deng, y., peng, s. and yan, s. - Positive soliton solutions for generalized quasilinear Schrodinger equations

with critical growth J. Differential Equations, 258, 115-147, 2015.

[3] li, q. and wu, x. - Existence, multiplicity and concentration of solutions for generalized quasilinear

Schrodinger equations with critical growth J. Math. Phys., 58, 30pp, 2017.

[4] severo, u. b. and germano, d. s. - Existence and nonexistence of solution for a class of quasilinear

Schrodinger equations with critical growth Acta Appl. Math., 2021. To appear.

[5] shen, y. and wang, y. - Soliton solutions for generalized quasilinear Schrodinger equations. Nonlinear Anal.,

80, 194-201, 2013.


ON THE FRACTIONAL P -LAPLACIAN CHOQUARD LOGARITHMIC EQUATION WITH

EXPONENTIAL CRITICAL GROWTH: EXISTENCE AND MULTIPLICITY

EDUARDO DE S. BOER1 & OLIMPIO H. MIYAGAKI2

1Departamento de Matematica, UFScar, SP, Brasil, [email protected],2Departamento de Matematica, UFScar, SP, Brasil, [email protected]

Abstract

In this work we present two main results concerning the existence and multiplicity of non-trivial solutions

for the Choquard logarithmic equation (−∆)spu+ |u|p−2u+(ln | · | ∗ |u|p)|u|p−2u = f(u) in RN , where N = sp,

s ∈ (0, 1), p > 2, a > 0, λ > 0 and f : R → R is a continuous nonlinearity with exponential critical growth.

Using variational techniques we guarantee the existence of a non-trivial solution at the mountain pass level

and a non-trivial ground state solution under critical growth. Moreover, via genus theory, considering f with

subcritical growth we prove the existence of infinitely many solutions.

1 Introduction

In the present work we are concerned with the existence and multiplicity of solutions to the following Choquard

logarithmic equation

(−∆)spu+ |u|p−2u+ (ln | · | ∗ |u|p)|u|p−2u = f(u) in RN , (1)

where N = sp, s ∈ (0, 1), p > 2, a = 1, λ = 1, f : R → R is continuous, with primitive F (t) =t∫0

f(τ)dτ , and (−∆)sp

is the fractional p-Laplacian operator.

The results presented here are based in [1] and inspired by [2], where the authors studied existence and

multiplicity results for the planar Schrodinger-Poisson system with polynomial nonlinearity, using variational

techniques and, when needed, considering subgroups of the rotational group O(2) ⊂ R2. For a more detailed

literature overview we refer to [1] and the references therein.

Based on works that deal with exponential growth nonlinearities, we ask the following conditions over f .

(f1) f ∈ C(R,R), f(0) = 0, has critical exponential growth and lim|t|→0

f(t)

|t|p−2t= 0.

(f2) there exists θ > 2p such that f(t)t ≥ θF (t) > 0, for all t > 0.

(f3) there exist q > 2p and Cq >[2(q − p)]

q−pp

qqp

Sqq

ρq−p0

such that F (t) ≥ Cq|t|q, for all t ∈ R, where Sq is a suitable

constant obtained from the Sobolev embeddings and ρ0 > 0 is a sufficiently small value.

Since the above conditions are sufficient to obtain the existence result, in order to get multiplicity we need a

stronger geometry for the energy functional. In this sense, we will need to consider the following.

(f ′1) f ∈ C(R,R), f is odd, has subcritical exponential growth and lim|t|→0

f(t)

|t|p−2t= 0.

(f ′3) there exists q > 2p and M1 > 0 such that F (t) ≥M1|t|q , ∀ t ∈ R,

(f4) the function t 7→ f(t)

t2p−1is increasing in (0,+∞).

The results presented here can be adapted to a general version of (P) considering a ZN -invariant (or

asymptotically ZN -invariant) continuous potential a : RN → R.

191

192

2 Main Results

In the following, we only sketchy the proof. Detailed arguments can be found in [1].

Theorem 2.1. Assume (f1) − (f3), q > 2p and Cq > 0 sufficiently large. Then,

(i) Problem (P) has a non-trivial solution u ∈ X such that

I(u) = cmp = infγ∈Γ

maxt∈[0,1]

I(γ(t)),

where Γ = γ ∈ C([0, 1], X) ; γ(0) , I(γ(1)) < 0 and X ⊂W s,p(RN ) is a Banach space.

(ii) Problem (P) has a non-trivial ground state solution u ∈ X, that is, u satisfies

I(u) = cg = infI(v) ; v ∈ X is a solution of (P).

Proof From conditions (f1)−(f3) it is possible to prove that I has the mountain pass geometry and, consequently,

there exists a Cerami sequence in the level cmp. Let (un) ⊂ X be such sequence. Then, as I(un) → cmp > 0, one

can prove that there exists a sequence (yn) ⊂ ZN such that, up to a subsequence, un = un(· − yn) → u in X for

a non-trivial critical value of I. Moreover, considering the set K = v ∈ X \ 0 ; I ′(v) = 0, that is not empty

by item (i). Hence, since cg ∈ [−∞, cmp], one can prove in an analogously way that a minimizing sequence for Kconverges to a critical point of I, u ∈ K, satisfying I(u) = cg.

Theorem 2.2. Suppose (f ′1), (f2), (f ′3) and (f4). Then, problem (P) has infinitely many solutions.

Proof From the hypothesis, we can prove that φu(t) = I(tu), for all u ∈ X and t ∈ (0,+∞) has the desired

geometry that allow us to guarantee that the values ck = infc ≥ 0 ; γD(Ic) ≥ k, for all k ∈ N, where D = I0,

Ic = u ∈ X ; I(u) ≤ c for c ∈ R and γD stands as the Krasnoselskii’s Genus relative to D, are critical values of

I and ck → +∞, as k → +∞.

References

[1] E. S. Boer and O. H. Miyagaki, Existence and multiplicity of solutions for the fractional p-Laplacian

Choquard logarithmic equation involving a nonlinearity with exponential critical and subcritical growth, J.

Math. Phys., 62, 051507, 2021.

[2] Du, M. and Weth, T. Ground states and high energy solutions of the planar Schrodinger-Poisson system.

Nonlinearity, 30, 3492-3515, 2017.


EQUACAO DE CHOQUARD: EXISTENCIA DE SOLUCOES DE ENERGIA MINIMA PARA UMA

CLASSE DE PROBLEMAS NAO LOCAIS ENVOLVENDO POTENCIAIS LIMITADOS OU

ILIMITADOS

EDUARDO DIAS LIMA1 & EDCARLOS DOMINGOS DA SILVA2

1Instituto de Matematica e Estatıstica, UFG, GO, Brasil, [email protected],2Instituto de Matematica e Estatıstica, UFG, GO, Brasil, [email protected]

Abstract

Neste trabalho, apresentamos um estudo sobre a existencia de solucao de energia mınima para a seguinte

equacao de Choquard nao linear−u+ V (x)u =

( ∫RN

Q(y)F (u(y))

|x− y|µ dy

)Q(x)f(u(x))

u ∈ D1,2(RN),

(1)

onde N ≥ 3, 0 < µ < N , V ∈ C(RN, [0,+∞)), Q ∈ C(RN, (0,+∞)), f ∈ C1(R,R) e F (t) =∫ t

0f(s)ds. A

nao-linearidade f : R → R e contınua e tem comportamento assintoticamente linear no infinito. Alem disso,

sobre certas condicoes da variedade de Nehari N e algumas outras desigualdades, estabelecidas no trabalho, a

equacao (1) tem uma solucao de energia mınima.

1 Introducao

Para a elaboracao deste trabalho, seguimos os artigos [1], [2] e [3]. Em 2018, os autores Sitong Chen e Shuai Yuan

estudaram a seguinte equacao de Choquard nao linear dado em (1). Consequentemente, expressaram o conjunto E

de modo que

E :=

u ∈ D1,2(RN) :

∫RNV (x)u2dx < +∞

,

afim de obter solucao fraca para (1), cujo objetivo e encontrar um ponto crıtico nao trivial para Φ. Por meio de

metodos variacionais, podemos definir o funcional energia natural associado ao problema (1), Φ : E → R por

Φ(u) =1

2

∫RN

|∇u|2dx+1

2

∫RN

V (x)u2dx− 1

2

∫RN

∫RN

Q(y)F (u(y))

|x− y|µdyQ(x)F (u(x))dx.

Mostramos que Φ e de classe C1(E,R). Recentemente, muitos pesquisadores comecaram a se concentrar na equacao

de Choquard com a nao linearidade nao homogenea satisfazendo as seguintes hipoteses:

(F0) f ∈ C(R,R) satisfaz

limt→0

F (t)

|t|(2N−µ)/N= 0 e lim

|t|→+∞

F (t)

|t|(2N−µ)/(N−2)= 0,

existe uma constante C0 > 0 tal que

|tf(t)| ≤ C0

(|t|(2N−µ)/N + |t|(2N−µ)/(N−2)

), ∀ t ∈ R.

(Q1) V (x), Q(x) > 0; ∀ x ∈ RN , V ∈ C(RN ,R) e Q ∈ C(RN ,R) ∩ L∞(RN ,R);

193

194

(Q2) Se An ⊂ RN e uma sequencia do conjunto de Borel tal que a medida de Lebesgue para An e menor do que

δ, ∀ n e algum δ > 0, entao

limr→+∞

∫An∩Bc

r(0)

[Q(x)]2N

2N−µ dx = 0, uniformemente em n ∈ N;

(Q3)Q

V∈ L∞(RN );

(Q4) Existe p ∈ (2, 2∗) tal que

[Q(x)]2N

2N−µ

[V (x)]2∗−p2∗−2

−→ 0, |x| → +∞.

(F1) lim|t|→+∞

F (t)

|t|= +∞;

(F2) limt→0

F (t)

|t| 2N−µN

= 0, se vale (Q3); ou limt→0

F (t)

|t|p(2N−µ)

2N

= 0, se vale (Q4).

(F3) lim|t|→+∞

F (t)

|t|2N−µN−2

< +∞, se vale (Q3); lim|t|→+∞

F (t)

|t|p(2N−µ)

2N

< +∞, se vale (Q4).

(F4) f(t) e nao-decrescente em R.

Neste trabalho, explicitamos a existencia de solucao de energia mınima por meio do conjunto de Nehari,

N := u ∈ E \ 0 : ⟨Φ′(u), u⟩ = 0.

Para isso, garantimos que o funcional Φ associado ao problema (1) possui a Geometria do Passo da Montanha,

utilizando as hipoteses de crescimento assumidas sobre a funcao f acima, e asseguramos a existencia de uma

sequencia limitada de Cerami (un)n∈N para Φ. Por fim, evidenciamos que o funcional Φ possui ınfimo e e atingido

para algum elemento u ∈ H1(RN ).

Note que (V,Q) ∈ K significa o conjunto de todos os potenciais V e Q tais que (Q1)-(Q4) sao satisfeitas.


O principal resultado deste trabalho pode ser descrito da seguinte forma:

Teorema 2.1. Suponha que (V,Q) ∈ K e f ∈ C1(R,R) satisfazendo (F1)-(F4). Entao (1) tem uma solucao de

energia mınima u ∈ E tal que Φ(u) = infN

Φ > 0.

References

[1] ALVES, C. O.; SOUTO, M. A. S. Existence of solutions for a class of nonlinear Schrodinger equations

with potential vanishing at infinity. J. Differential Equations, 2013.

[2] CHEN, S.; YUAN, S. Ground state solutions for a class of Choquard equations with potential

vanishing at infinity, J. Math. Appl. 463: 880-894, 2018.

[3] MOROZ, V.; SCHAFTINGEN, J. V. Existence of groundstate for a class of nonlinear Choquard

equations, Trans. Amer. Math. Soc. 367: 6557-6579, 2015.


FULLY NONLINEAR SINGULARLY PERTURBED MODELS WITH NON-HOMOGENEOUS

DEGENERACY

ELZON C. JUNIOR1, V2 & JOAO VITOR DA SILVA3

1Depto. de Matematica, UFC, CE, Brasil, [email protected],2Depto. de Matematica, UFC, CE, Brasil, [email protected],


Abstract

This work is devoted to studying non-variational, nonlinear singularly perturbed elliptic model enjoying a

double degeneracy character with prescribed boundary value domain. For each ϵ > 0 fixed, we seek a non-

negative function uϵ satisfying[|∇uε|p + a(x)|∇uε|q]∆uε = ζϵ(x, u

ϵ) em Ω

uε(x) = g(x) em ∂Ω,

in the viscosity sense for suitable p, q ∈ (0,∞), a, g, where ζϵ one behaves singularly of order O(ε−1) near

ε-surfaces. In such context, we establish that solutions are locally (uniformly) Lipschitz continuous, and they

grow in a linear fashion.

Keywords: Singular perturbation methods, doubly degenerate fully non-linear operators, geometric regularity

theory.

1 Introduction

In this work we shall develop to study (locally) sharp and geometric estimates of one-phase solutions to a singularly

perturbed problem having a non-homogeneous degeneracy, whose mathemtical model is given by: Fixed a parameter

ε ∈ (0, 1), we would like to find uε ≥ 0 viscosity solution to[|∇uε|p + a(x)|∇uε|q]∆uε = ζϵ(x, u

ϵ) in Ω

uε(x) = g(x) on ∂Ω,(1)

for a bounded domain Ω ⊂ Rn, where 0 ≤ g ∈ C0(∂Ω).

In a few words, under appropriated hypothesis on data, we show that, for ε→ 0+, the family uεε>0 to (1) are

asymptotic approximations to a one-phase u0 of an inhomogeneous non-linear free boundary problem, which arises

in the mathematical formulation of some issues in flame propagation and combustion theory.

We suppose that the expoents p, q and the modulating function a(·) fulfil

0 < p ≤ q <∞ e a ∈ C0(Ω, [0,∞)). (2)

The reaction term, i.e, ζϵ : Ω × R+ → R+ , represents the singular perturbation of the model. In this point, we

are interested in a singular behaviour or order O(1ε

)along uϵ ∼ ϵ. Hence, we are led to consider reaction terms

fulfilling

B0 ≤ ζϵ(x, t) ≤Aϵχ(0,ϵ)(t) + B, ∀ (x, t) ∈ Ω × R+, (3)

195

196

for nonnegative constants A, B0, B ≥ 0. Note that ζϵ ≡ 0 satisfies (3). Then, we shall also impose the following

non-degeneracy assumption in order to ensure that such a reaction term enjoys an authentic singular character:

I = infΩ×[t0,T0]

ϵζϵ(x, ϵt) > 0, (4)

for some constants 0 ≤ t0 < T0 <∞, where I does not depend on ϵ.

2 Main Results

Teorema 2.1 (Optimal Lipschitz estimate ). Let uϵϵ>0 be a solution(1). Dado Ω′ ⋐ Ω, there exists a

constant C0 depending on dimension and on Ω′, but independet of ϵ > 0, such that

∥∇uϵ∥L∞(Ω′) ≤ C0.

Additionaly, if uϵϵ>0 is a uniformly bounded family, 1, then it is pre-compact in the Lipschitz topology

From now on, we will label the distance of a point in the non-coincidence set x0 ∈ Ω ∩ uϵ > 0 to the

approximation boundary, Γϵ, pby

dϵ(x0) = dist(x0, uϵ ≤ ϵ).

Teorema 2.2 (Linear growth). Let uϵϵ>0 be a Perron’s solution to(1). There exists a positive constant

c( parameters) > 0 such that, for x0 ∈ uϵ > ϵ and 0 < ϵ≪ dϵ(x0) ≪ 1, there holds

uϵ(x0) ≥ c · dϵ(x0).

References

[1] D.J. Araujo, G.C. Ricarte, E.V. Teixeira, Singularly perturbed equations of degenerate type. Ann. Inst. H.

Poincare Anal. Non Lineaire 34 (2017), no. 3, 655-678.

[2] G.C. Ricarte and J.V. Silva, Regularity up to the boundary for singularly perturbed fully nonlinear elliptic

equations, Interfaces and Free Bound. 17 (2015), 317-332.

[3] G.C. Ricarte and E.V. Teixeira, Fully nonlinear singularly perturbed equations and asymptotic free boundaries.

J. Funct. Anal. 261 (2011), no. 6, 1624–1673

1Such a bound will be universal, i.e, it will depend only on data of the problem. Moreover, this statement is obtained via the

application of Aleandroff-Bakelmann-Pucci estimate adapted to our context.


A GEOMETRIC APPROACH TO INFINITY LAPLACIAN WITH SINGULAR ABSORPTIONS

GINALDO S. SA1 & DAMIAO J. ARAUJO2

1Departamento de Matematica, UFPB, Brasil, [email protected],2Departamento de Matematica, UFPB, Brasil, [email protected]

Abstract

This work is dedicated to the study of a nonvational class of singular elliptic equations ruled by the infinity

Laplacian. We provide optimal regularity C1,β regularity along the singular free boundary, where in particular,

the optimality is designed by the magnitude of the singularity. By virtue of a suitable penalization scheme

and a construction of radial super-solutions, existence and non-degeneracy properties are provided where fine

geometric consequences for the singular free boundary are further obtained.

1 Introduction

The main purpose of this work is to study geometric and qualitative properties for nonnegative viscosity solutions

of the following singular free boundary problem∆∞u = u−γ in Ω ∩ u > 0,u = φ on ∂Ω,

(1)

with 0 ≤ γ < 1,Ω ⊂ Rn is a bounded smooth domain and φ ≥ 0 is a given smooth boundary data. This type of

singular free boundary problem brings extra difficulties. In this case, the nonhomogeneity term blows up along the

a priori unknown set ∂u > 0− called free boundary. In order to circumvent this issue, we shall deal with the

penalized problem ∆∞u = Bε(u)u−γ in u > 0,u = φε on ∂Ω,

where the term Bε(s) is a suitable approximation for the function χs>0 , and for each parameter ε > 0 the source

term Bε(s)s−γ is a Lipschitz function defined on R. In addition, the boundary data φε is assumed approximating

φ given in (1).

2 Main Results

In view of this, we shall obtain regularity estimates for limiting solutions u of (1), given by

u = limε→0

uε

Nonetheless, this approach still brings a delicate issue: since nontrivial nonnegative solutions uε may not present

free boundaries ∂uε > 0, it would not be reasonable to get estimates at the free boundary ∂u > 0 as limits of

the ones at ∂uε > 0. This is well observed in the radial examples that satisfy

infx∈Ω

uε(x) ≳ ε.

In order to overcome this problem, we prove the main ingredient of our analysis: a new oscilation estimate at

floating level sets. The innovative feature concerns an interrelation between radii and appropriate nonzero level

sets. More specifically, we prove the following theorem:

197

198

Theorem 2.1 (Optimal oscillation estimates for floating level sets). There exist constants C and κ0

depending on universal parameters, with no dependence on ε > 0, such that if v is a nonnegative viscosity solution

of

∆∞v = Bε(v)v−γ em B1

then, for any

v(x)1α ≤ κ ≤ κ0,

there holds

supBκ(x)

v ≤ Cκα,

for the exponent

α =4

3 + γ.

Thanks to the Theorem (??), it was possible to prove that at points on the free boundary ∂u > 0, limiting

solutions are precisely of the class C1,1−α . Surprisingly, such regularity is essentially superior than regularity

results involving nonsingular infinity Laplacian equations, even considering the class of infinity harmonic functions.

Consequently, this result allowed us to obtain estimates of non-degeneracy and geometric measurement property

for the free boundary ∂u > 0.

References

[1] D.J. Araujo, R. Leitao, E.V. Teixeira - Infinity Laplacian equation with strong absorptions, J. Funct.

Analysis 270 (2016), 2249-2267.

[2] D.J. Araujo, E.V. Teixeira - Geometric approach to nonvariational singular elliptic equations, Arch.

Ration. Mech. Anal. 209 (2013), 1019-1054.

[3] G. Aronsson - Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551-561.

[4] E. Lindgren - On the regularity of solutions of the inhomogeneous infinity Laplace equation, Proc. Amer.

Math. Soc. 142 (2014), 277-288.

[5] G. Lu, P. Wang - Inhomogeneous infinity Laplace equation, Adv. Math. 217 (2008), 1838-1868.


EXISTENCIA DE SOLUCOES POSITIVAS PARA O P-LAPLACIANO FRACIONARIO

ENVOLVENDO NAO LINEARIDADE CONCAVO CONVEXA

JEFFERSON LUIS ARRUDA OLIVEIRA1 & EDCARLOS DOMINGOS DA SILVA2

1Instituto de Matematica e Estatıstica, UFG, GO, Brasil, [email protected],2Instituto de Matematica e Estatıstica, UFG, GO, Brasil, [email protected]

Abstract

Neste trabalho, estabelecemos a existencia de solucoes positivas para o problema elıptico com nao linearidades

do tipo concavo-convexa, dado por(−∆)spu + V (εx) |u|p−2 u = λf(εx) |u|q−2 u + g(εx) |u|r−2 u, em RN ,

u ∈W s,p(RN ).(Pε)

onde ε, λ > 0 sao parametros positivos, N > ps com s ∈ (0, 1) fixado, 1 < q < p < r < p∗s , e p∗s = Np

N−ps.

Consideramos hipoteses adequadas sobre as funcoes f e g, e para o potencial V , para concluir um resultado

de existencia de solucoes positivas para o problema acima utilizando a conhecida variedade de Nehari. Mais

especificamente demonstramos a existencia de uma solucao ground state positiva em N+ε e outra solucao positiva

em N−ε .

1 Introducao

Este trabalho e um recorte do trabalho final de dissertacao o qual foi motivado pelo artigo dos autores Qingjun

Lou e Hua Luo, [1]. Para demonstrarmos o resultado de existencia de solucoes para o problema (1), consideramos

as seguintes hipoteses sobre as funcoes f e g:

(F ) f ≥ 0, ≡ 0, f ∈ Lq(RN ) ∩ C(RN ), (q =r

r − q) onde |f |q > 0 e

fmax := maxx∈RN

f(x) = 1;

(G) g e uma funcao contınua, positiva e definida em RN . Alem disso, g(x) ≤ 1 para todo x ∈ RN .

Para o potencial V consideramos a seguinte hipotese:

(V) V ∈ C(RN ,R) e satisfaz

V∞ := lim inf|x|→+∞

V (x) > V0 := infx∈RN

V (x) > 0.

A hipotese (V ) e muito comum em trabalhos dessa natureza e foi introduzida em [3] por Rabinowitz. Neste trabalho

Rabinowitz demonstrou que se o potencial e coercivo, e possıvel garantir a existencia de imersoes compactas do

espaco de trabalho para o espaco Lt(RN ), com t ∈ [2, 2∗), onde 2∗ = 2NN−2 , mesmo trabalhando sobre um domınio

ilimitado. Ressaltamos que este resultado nao se aplica em nosso caso, pois existem potenciais satisfazendo (V ) que

nao sao coercivos. Portanto a perda da compacidade foi um dos principais problemas abordados.

Ao problema (1) associamos o seguinte funcional energia

Iε(u) =1

p

∫RN

∫RN

|u (x) − u (y)|p

|x− y|N+spdxdy+

1

p

∫RN

V (εx) |u (x)|p dx− λ

q

∫RN

f (εx) |u (x)|q dx− 1

r

∫RN

g (εx) |u (x)|r dx.

199

200

Uma vez que estamos com potenciais gerais, afim de recuperar algumas propriedades importantes, definimos o

seguinte espaco de trabalho:

Xε =

u ∈W s,p(RN ) :

∫RN

V (εx) |u|p dx <∞.

Neste momento ressaltamos que as principais informacoes sobre o operador p-Laplaciano fracionario e o espaco

de Sobolev fracionario W s,p(RN ), foram obtidas de um modo geral por [2]. Para iniciarmos a construcao do

resultado, foi necessario definirmos as fibras ligadas ao funcional Iε as quais podem ser escritas do seguinte modo,

γu : R+∗ −→ R e para cada funcao u fixada, temos que γu(t) = Iε(tu). Na sequencia introduzimos a famosa

variedade de Nehari, dada por

Nε = u ∈ Xε \ 0 : γ′u(1) = 0 .

Para alcancarmos nosso resultado, dividimos a variedade acima em tres novos conjuntos,

N+ε =

u ∈ Nε; γ

′′

u (1) > 0,

N−ε =

u ∈ Nε; γ

′′

u (1) < 0,

N 0ε =

u ∈ Nε; γ

′′

u (1) = 0,

e exibimos condicoes suficientes para que N 0ε seja vazia. Deste modo, utilizando a coercividade do funcional Iε

sobre Nε, fomos capazes de demonstrar a existencia de uma sequencia de Palais-Smale em cada uma das variedades

N+ε e N−

ε , as quais nos forneceram um resultado de existencia de solucao para o problema (1). Na sequencia,

regularizamos estas solucoes via iteracao de Moser e por fim aplicarmos o princıpio do maximo forte de onde

estabelecemos a existencia de solucoes positivas para o problema (1).


O principal resultado deste trabalho pode ser descrito da seguinte forma:

Teorema 2.1. Seja 0 < λ < qpλ0, onde λ0 e um parametro suficientemente pequeno. Suponha ainda que f, g e V

satisfacam as condicoes (F ), (G), e (V ). Entao o problema (1) possui pelo menos 2 solucoes positivas, uma em

N+ε e outra em N−

ε .

References

[1] LOU, Qingjun; LUO, Hua. Multiplicity and concentration of positive solutions for fractional p-Laplacian

problem involving concave-convex nonlinearity, Nonlinear Analysis: Real World Applications, v. 42, p. 387-408,

2018.

[2] DI NEZZA, Eleonora; PALATUCCI, Giampiero; VALDINOCI, Enrico. Hitchhiker’ s guide to the fractional

Sobolev spaces, Bulletin des sciences mathematiques, v. 136, n. 5, p. 521-573, 2012.

[3] RABINOWITZ, Paul H. On a class of nonlinear Schrodinger equations, Zeitschrift fur angewandte Mathematik

und Physik ZAMP, v. 43, n. 2, p. 270-291, 1992.


STABILIZED HYBRID FINITE ELEMENT METHODS FOR THE HELMHOLTZ PROBLEM

MARTHA H. T, SANCHEZ1 & ABIMAEL F. D. LOULA2

1Institute of Investigation of the Faculty of Mathematical Sciences, UNMSM, Lima, Peru and National Laboratory for

Scientific Computing, Petropolis, Brasil, [email protected]@lncc.br,2National Laboratory for Scientific Computing, Petropolis, Brasil, [email protected].

Abstract

1 Stabilized hybrid finite element methods are proposed for the Helmholtz problem with Robin’s condition

using two different types of multipliers. These multipliers (continuous or discontinuous) are introduced to

weakly impose continuity at the interfaces of the finite elements. We presented numerical results illustrating the

great stability , precision and robustness of these formulations adopting polynomial spaces for the pressure and

multipliers.

1 Introduction

Acoustic waves (sound) are small pressure fluctuations in an understandable fluid. These oscillations interact in

such a way that the energy spreads through the medium. Assuming a linear constitutive law and considering

the propagation of harmonic waves over time, we obtain the Helmholtz equation whose solutions depend on a

parameter κ, called wave number [1], which characterizes the frequency of oscillations of harmonic solutions. As

analyzed by Ihlenburg and Babuska [5], the finite element method with linear approximations presents adequate

asymptotic behavior, with optimal convergence rates, only for extremely refined meshes, which obey the condition

κ2h ≤ 1, which makes this approach unviable for real problems with high numbers of waves κ. Loula and

Fernandez [6] proposed a Petrov-Galerkin (QOPG) method whose weight functions are obtained by minimizing

a local least squares functional truncation error. This method has good properties of stability, precision, generality

and robustness.

To determine better approximations methods of discontinuous finite elements (DG) have been proposed. Despite

the advantages offered by the DG methods, due to its formulation complexity, computational implementation and

a high number of degrees of freedom have been proposed hybridizations for the DG methods in order to derive

new finite element methods with better stability characteristics and reduced computational cost but preserving the

robustness and flexibility of the DG Methods, [2].

We will study the Helmholtz equation

− ∆p− κ2p = f, em Ω (1)

with Robin’s condition,

−∇p · n + iκp = g, em ∂Ω (2)

where Ω ⊂ R2 is a polygonal domain. We present three methods stabilized hybrid finite element methods, two

with Lagrange multiplier associated with pressure, one with continuous multiplier denoted LDGC-P and another

discontinuous denoted LDGD-P, and the third with multiplier associated with speed denoted LDGF-P.

1This research was supported by the CNPq and Universidad Nacional Mayor de San Marcos - RR No 005753-2021 and project

number B21142201.

201

202

2 Main Results

The obtained approximations present reduced numerical pollution, optimal rates of convergence, flexibility and

robustness. Numerical studies have shown that with the same number of global degrees of freedom, the LDGC-P

method is more accurate than the LDGD-P method. In the case of the LDGD-P method projected, the choices

Q2 − p1 and P2 − p1 can recover the optimal convergence rates for the primal variable. It was also observed that

this projection applied to the LDGC-P method, with a continuous multiplier, does not have the same stability

and precision as the LDGD-P method, particularly with triangular elements. We also analyzed a stabilized

primal hybrid formulation LDGD-F, where the multiplier is associated with the flow, as in the classic primal

hybrid formulation of Raviart and Thomas [9] and in its stabilized version proposed by Ewing, Wang and Yang [3].

Compared with the formulation LDGD-P we can see that, choosing the degree s of the polynomial approximation

of the multipliers equal or greater than the degree l of the polynomial approximation of the primal variable, that is

s ≥ l, the hybrid methods LDGD-P and LDGD-F provide the same approximations for the primal variable ph.

However, for s = l − 1 these approaches differ. Results of convergence studies show that, for this choice s = l − 1,

the LDGD-P method can present optimal convergence rates for the primal variable ph when is used a projection

of the terms of edges in the multiplier spaces. Already the LDGD-F method presented optimal convergence rates

of the primal variable ph, for the choice s = l − 1, without the need of this projection.

References

[1] ARRUDA, N. C. B.; LOULA, A. F. D.; ALMEIDA, R. C. Locally discontinuos but globally continuos

Galerkin methods for elliptic problems., v. 255, p. 104–120, 2013.

[2] COCKBURN, J. G. B. and LAZAROV, R. Unified hybridization of discontinuos Galerkin, mixed, and

continuous Galerkin methods for second order elliptic problems., J. Numer. Anal., v. 47, n. 2, p. 1319–1365,

2009.

[3] EWING, R. E.; WANG, J. and YANG, Y. A stabilized discontinuous finite element method for elliptic

problems., Numer. Linear Algebra Appl., v. 10, p. 83–104, 2003.

[4] IHLENBURG, F. Finite Element Analysis of Acoustic Scattering., Springer, 1998.

[5] IHLENBURG, F. and BABUSKA I. Finite element solution of the Helmholtz equation with high wave

number part i: The h-version of the fem., Computer Math. Applic, v. 30, n. 9, p. 9–37, 1995.

[6] LOULA, A. F. D. and FERNANDES, D. T. A quasi optimal Petrov Galerkin method for Helmholtz

problem., International Journal for Numerical Methods in Engineering, v. 80, p. 1595–1622, 2009.

[7] LOULA, A. F. D.; ALVAREZ, G. B.; DO CARMO, E. G. D. A discontinuous finite element method at

element level for Helmholtz equation., Computer methods applied mechanics and enginnering-Elsevier Science,

v. 196, p. 867–878, 2007.

[8] NUNEZ, Y. R.; FARIA, C. O.; LOULA, A.F. D. and MALTA, S.M. C. Um metodo hıbrido de elementos

finitos aplicado a deslocamentos miscıveis em meios porosos heterogeneos., Revista Internacional de Metodos

Numericos para Calculo y Diseno en Ingenierıa, v. 33, p. 45–51, 2017.

[9] RAVIART, P. A. and THOMAS, J. M. Primal hybrid finite element methods for 2nd order elliptic

equations., Mathematics of Computation, v. 33, p. 391–413, 1977.


PROBLEMA QUASELINEAR DE AUTOVALOR COM NAO-LINEARIDADE DESCONTINUA

J. ABRANTES SANTOS1, PEDRO FELLYPE S. PONTES2 & SERGIO HENRIQUE M. SOARES3

1Departamento de Matematica, UFCG, PB, Brasil, [email protected]; Apoiado por CNPq/Brazil 303479/2019-1,2Departamento de Matematica, UFCG, PB, Brasil, [email protected],

3ICMC, USP, SP, Brasil, [email protected]

Abstract

Neste trabalho apresentaremos um resultado de existencia e nao-existencia de solucoes positivas para uma

classe de problemas quaselineares de autovalor. Mostramos a existencia de uma aplicacao contınua Λ : [0, a⋆] → Rtal que o grafico dessa aplicacao define a regiao de existencia e nao-existencia de solucoes positivas. A ferramenta

principal usada sao os metodos variacionais para funcionais localmente Lipschitz nos espacos de Orlicz-Sobolev.

1 Introducao

Neste presente trabalho estamos interessados em solucoes positivas para a seguinte classe de problemas quaselineares−∆Φu = λf(x, u)χ[u≥a] em Ω,

u = 0 sobre ∂Ω,(1)

onde Ω ⊂ RN e um domınio limitado, N ≥ 2, a e λ sao parametros prositivos, χ e a funcao caracterıstica, f e uma

funcao contınua satisfazendo condicoes apropriadas e Φ : R → R e uma N-funcao dada por Φ(t) =∫ |t|0sϕ(s)ds,

onde ϕ : (0,+∞) → (0,+∞) e uma funcao de classe C1. Com o intuito de utilizar metodos variacionais, inspirados

por Fukagai e Narukawa [2], assumimos que ϕ satisfaca as seguintes condicoes:

(ϕ1) (ϕ(t)t)′ > 0, t > 0;

(ϕ2) existem l,m ∈ (1, N) com m ∈ [l, l∗) e l∗ = lNN−l , tais que

l ≤ Φ′(t)t

Φ(t)≤ m t > 0;

(ϕ3) existem k0, k1 > 0 tais que

k0 ≤ Φ′′(t)t

Φ′(t)≤ k1 t > 0;

(ϕ4) ϕ e uma funcao monotona nao-decrescente em (0,∞);

Da mesma forma, assumiremos que f : Ω × R → R e uma funcao contınua verificando:

(f1) para x ∈ Ω, f(x, t) ≤ 0 para t ≤ 0, e f(x, t) > 0 para t > 0;

(f2) existe C0 > 0 tal que

f(x, t)t ≤ C0Φ(t);

(f3) limt→+∞

f(x, t)t

Φ(t)= 0, uniformemente em Ω;

(f4) limt→0

f(x, t)t

Φ(t)= 0, uniformemente em Ω;

203

204

(f5) para cada x ∈ Ω, a funcao f(x, ·) e nao-decrescente em R+.

Observamos que o problema (1), para o caso a = 0, e exatamente o estudado por Fukagai e Narukawa em

[2]. Contudo, para a > 0 a existencia de solucoes positivas nao e tao simples, pois estamos com nao-linearidade

descontınua, logo temos que trabalhar com a teoria de funcionais Lipschitz e gradientes generalizados.

Enfatizamos que entendemos por solucao do problema (1), uma funcao uλ,a ∈W 1,Φ0 (Ω) satisfazendo:

i) |[uλ,a ≥ a]| > 0;

ii) existe ζ(·, uλ,a) ∈ LΦ(Ω) tal que∫Ω

ϕ(|∇uλ,a|)∇uλ,a∇vdx = λ

∫Ω

ζ(x, uλ,a)vdx, v ∈W 1,Φ0 (Ω).

Alem disso, ζ(x, uλ,a) ∈ ∂Fa(x, uλ,a) q.t.p. x ∈ Ω, onde Fa(x, t) =∫ t

0χ[s≥a]f(x, s)ds.

Ademais, temos um segundo sentido de solucao, o qual e apoiada por Gasinski e Papageorgiou em [1]. Dizemos

que uλ,a e uma S-solucao para o problema (1) se, uλ,a e solucao de (1) e |[uλ,a = a]| = 0.


Teorema 2.1. Suponha que ϕ e f sao funcoes satisfazendo (ϕ1)-(ϕ4) e (f1)-(f5), respectivamente. Existem uma

constante a⋆ > 0 e uma aplicacao contınua nao-decrescente Λ : [0, a⋆] → R+, tais que, para cada a ∈ [0, a⋆]:

i) para todo λ ∈ (0,Λ(a)), o problema (1) nao possui solucao;

ii) para λ = Λ(a), o problema (1) possui pelo menos uma S-solucao positiva;

iii) para todo λ > Λ(a), o problema (1) possui pelo menos duas solucoes positivas uλ,a e vλ,a satisfazendo vλ,a ≤ uλ,a

e vλ,a = uλ,a em Ω, onde uλ,a e uma S-solucao.

Prova: Inicialmente mostramos que para cada a⋆ > 0 existe λ⋆ := λ(a⋆) > 0 (de modo que se a⋆ → +∞, entao

λ⋆ → +∞) tal que para todo (a, λ) ∈ [0, a⋆]× [λ⋆,+∞) o problema (1) possui uma S-solucao positiva em C1,α0 (Ω),

para algum α ∈ (0, 1). Dessa forma, podemos definir a aplicacao Λ : [0, a⋆] → R+ dada por

Λ(a) := infλ ∈ R+ : existe uma S-solucao positiva de (1).

Assim, o item (i) fica mostrado. Para os outros itens aplicamos os metodos de sub e supersolucao e Teorema

do Passo da Montanha para funcionais localmente Lipschitz ao funcional

Iλ,a(u) =

∫Ω

Φ(|∇u|)dx− λ

∫Ω

Fa(x, u)dx, u ∈W 1,Φ0 (Ω).

References

[1] L. Gasinski & N. S. Papageorgiou, Nonsmooth critical point theory and nonlinear boundary value problems,

Series in mathematical analysis and applications ; v. 8. Boca Raton: Chapman & Hall/CRC, 2005.

[2] N. Fukagai & K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue

problems, Annali di Matematica (2007) 186(3):539?564


MULTIPLICITY OF SOLUTIONS TO A SCHRODINGER PROBLEM WITH SQUARE DIFFUSION

TERM

CARLOS ALBERTO SANTOS1, KAYE SILVA2 & STEFFANIO MORENO3

1Departamento de Matematica Universidade de Brasılia, Brasil, [email protected],2Universidade Federal de Goias, Goiania, GO, Brazil, [email protected],

3Universidade Federal de Goias, Goiania, GO, Brazil, [email protected]

Abstract

In this paper we show multiplicity of solutions for a parameterized quasilinear Schrodinger equation in the

presence of a square diffusion and indefinite superlinear term. Due to the presence of the quasilinear term,

we can no longer work on the standard Sobolev spaces to show existence and non-existence of solutions. We

overcome these difficulties by using perturbations arguments, Nehari sets and nonlinear Rayleigh quotients. As

a by product of this approach we show that the associated energy functional has non-zero global minimizers

only for small parameters.

1 Introduction

This work is concerned mainly with existence and multiplicity of solutions for the quasilinear Schrodinger equation−∆u− κ

2u∆u2 = f(x)|u|p−2u in Ω,

u = 0 on ∂Ω,(1)

where ∆ is the Laplacian operator, κ > 0, p ∈ (2, 4), f ∈ L∞(Ω) may change its sign, and Ω ⊂ RN is a smooth

bounded domain.

Due to the presence of the square diffusion term u∆u2, it is well known that the natural functional space

H10 (Ω) := W 1,2

0 (Ω) is “too big” to look for variational solutions, while W 1,40 (Ω) may be “too small” and so a natural

candidate would be the metric, but not vector space,

X = u ∈ H10 (Ω) :

∫u2|∇u|2 <∞

endowed with distance function given by

dX(u, v) := ∥u− v∥1,2 + ∥∇u2 −∇v2∥2.

Even though one makes sense to define a function u ∈ X as a weak solution of (1) whenever∫(1 + κu2)∇u∇φ+ κ

∫u|∇u|2φ =

∫f(x)|u|p−2uφ

holds for all φ ∈ C∞0 (Ω), the lack of closedness of X with respect to its metric dX leads X to be also“too big” to

approach the problem (1) in a variational sense. So, after these points, we were led to infer that the framework

Y := (Y, dX), defined by

Y = W 1,40 (Ω)

dX

.

205

206

In spite of Y seems to be an appropriate space, we have no guarantee that it is a linear normed space, which

prevents us to apply directly the usual minimax techniques to the energy functional

Φκ(u) =1

2

∫(1 + κu2)|∇u|2 − 1

p

∫f |u|p, u ∈ Y,

to find its critical points (that we call weak solutions of (1)), that is, functions u ∈ Y such that Φ′κ(u)φ = 0 for all

φ ∈ C∞0 (Ω), where

Φ′κ(u)φ =

∫(1 + κu2)∇u∇φ+ κ

∫u|∇u|2φ−

∫f |u|p−2uφ, φ ∈ C∞

0 (Ω).

Inspired on ideas from [1, 3, 2], we approach our problem by using a perturbation of the original energy functional

Φκ, defined by

Iµ,κ(u) :=µ

4

∫|∇u|4dx+ Φκ(u), u ∈W 1,4

0 (Ω).

2 Main Results

Theorem 2.1. Let p ∈ (2, 4). Then:

(i) the problem (1) admits two solutions wκ, uκ ∈ Y ∩ L∞(Ω), for each κ ∈ (0, κ∗0), that satisfy Φκ(wκ) > 0 and

infu∈Y Φκ = Φκ(uκ) < 0. Moreover, ∥uκ∥ → ∞ as κ→ 0,

(ii) the problem (1), for κ = κ∗0, admits two solutions wκ∗0, uκ∗

0∈ Y ∩ L∞(Ω) that satisfy Φκ∗

0(wκ∗

0) > 0 and

infu∈Y Φκ∗0

= Φκ∗0(uκ∗

0) = 0,

(iii) if κ > κ∗0, then infu∈Y Φκ = 0 and u = 0 is the only minimizer. Moreover, the problem (1) does not admits

any non-trivial solution for any κ > κ∗ .

Moreover, wκ, uκ, for κ ∈ (0, κ∗0), are bifurcations-solutions for the solutions wµ,κ, uµ,κ of Problem (??) at µ = 0.

References

[1] X.-Q. Liu, J.-Q. Liu, Z.-Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. of the A.M.S.

141 (2013), 253-263.

[2] X.-Q. Liu, J.-Q. Liu, Z.-Q. Wang,Quasilinear elliptic equations with critical growth via perturbation method. J.

Differential Equations 254 (2013), 102-124.

[3] J.-Q. Liu, X.-Q. Liu, Z.-Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via

perturbation method. Comm. Partial Differential Equations 39 (2014), 2216-2239.


O PROBLEMA DE DIRICHLET PARA UMA CLASSE DE EQUACOES DO TIPO P-LAPLACIANO

GLECE VALERIO KERCHINER1 & WILLIAM S. DE MATOS2

1Instituto de Matematica e Estatistica, UFRGS, RS, Brasil, [email protected],2Universidade Tecnologica Federal do Parana, PR, Brasil, [email protected]

Abstract

Neste trabalho, estudamos o problema de Dirichlet para a seguinte equacao diferencial parcial−div

(a(|∇u|)|∇u| ∇u

)= F (x, u) em Ω

u = g em ∂Ω,

onde Ω e um dominio limitado de classe C2,α contido em uma variedade Riemanniana completaM , g ∈ C2,α(Ω),

F : Ω× R → R e a : [0,+∞) → R sao funcoes satisfazendo determinadas condicoes.

1 Introducao

Seja M uma variedade rimanniana completa e Ω ⊂ M um dominio limitado de classe C2,α. Consideremos o

problema de dirichlet

(P.D) =

−Q(u) = F (x, u) em Ω

u = g em ∂Ω

onde g ∈ C2,α(Ω), Q(u) = div(

a(|∇u|)|∇u| ∇u

), a : [0,+∞) → R e tal que a ∈ C([0,+∞)) ∩ C1((0,+∞)), a > 0 e

a′ > 0 em (0,+∞) e a(0) = 0. Para garantir a elipticidade e exigido conforme [1] que

min0≤s≤s0

A(s), 1 +

sA′(s)

A(s)

> 0

para todo s0 > 0, onde escrevemos a(s) = sA(s).

Alem disso supomos que F : Ω ×R → R e nao-crescente em t ∈ R. Notemos que quando a(s) = sp−1, p > 1, temos

que Q(u) = div(|∇u|p−2∇u

)que e o operador do p-laplaciano.

O problema de dirichlet acima e uma generalizacao do caso em que F=0. Os autores em [1] estudam esse caso

particular. Muitos resultados obtidos em [1] se estendem para o caso F = 0. Dentre eles, destacamos o resultado

que segue abaixo.


Teorema 2.1. Fixemos p ∈ M . Seja Ω dominio limitado de classe C2,α,Ω compacto e suponhamos que

Ω ⊂ M − Cm(p) ∪ p, onde Cm(p) e o lugar dos pontos minimos. Suponhamos tambem que g ∈ C2,α(Ω).

Suponhamos que F : Ω × R → R de classe C1 e tal que Ft(x, t) ≤ 0, ∀(x, t) ∈ Ω × R. Denotemos a(s) = sA(s) e

assumimos que

(i) A ∈ C1,α ([0,∞)) ∩ C2,α ((0,∞)) ,

min0≤s≤s0

A(s), 1 +

sA′(s)

A(s)

> 0

para todo s0 > 0.

207

208

(ii) Existe uma funcao nao-decrescente φ : [s0,+∞) → R para algum s0 > 0 tal que∫ +∞

s0

φ(s)

s2dx = +∞

e

(1 + b−)s2 ≥ φ(s)

onde b(s) = sa′(s)a(s) − 1 e b−(s) = minb, 0.

(iii) Existem α0 > 0 e α1 > 0 tal que

α1 ≥ 1 + b(s) ≥ α0, ∀s ≥ 0.

(iv) Existem s0 > 0, β > 0 e uma funcao ψ ∈ C([0,+∞)) com lims→+∞

ψ(s) = +∞ tal que

(b(s) + 1 − βb′+(s)s− β(b(s) + 1)2

)s2 ≥ ψ(s), ∀s ≥ s0

onde b′+(s)= maxb′(s), 0.

Entao o problema de Dirichlet, tem uma unica solucao u ∈ C2,α(Ω).

References

[1] Rippol, j. b. and Tomi, f. - Notes on the dirichlet problem of a class of second order elliptic partial diferential

equations on a riemannian manifold; Ensaios Matematicos Sociedade Brasileira de Matematica., Rio de Janeiro,

volume 32, 2018.

[2] Carmo, M.do. - Geometria Riemanniana 5.ed. Rio de Janeiro: Projeto Euclides, 2015.

[3] gilbarg, d. and trudinger, n. s. - Elliptic Partial Differential Equations of Second Order. Berlim: Springer-

Verlag, 2001.


EXISTENCIA E REGULARIDADE PARA A SOLUCAO DE UM SISTEMA MULTIFASICO DA

ELETROHIDRODINAMICA

ANDRE F. PEREIRA1

1CEFET-MG, MG, Brasil, [email protected]

Abstract

Apresentaremos a analise matematica de um sistema de equacoes diferenciais parciais que modela um fluido

multifasico sob o efeito de um campo eletrico. Provamos a existencia de solucao fraca global e mostramos

resultados de regularidade, global no caso bidimensional e local no caso tridimensional.

1 Introducao

Estudamos o seguinte sistema de equacoes:

∂u

∂t+ (u · ∇)u = −∇p+ 2div(µ(c)D(u)) + ρT (ρ) + ϕ∇c em QT , (1)

div(u) = 0 em QT , (2)

∂ρ

∂t+ (u · ∇)ρ+ ρ = 0 em QT , (3)

∂c

∂t+ (u · ∇)c = div(M(c)∇ϕ) em QT , (4)

ϕ = Ψ′(c) − ∆c em QT , (5)

u =∂c

∂n=∂ρ

∂n=∂∆c

∂n= 0 sobre ∂Ω × (0, T ), (6)∫

Ω

ρ = 0, (7)

u(0) = u0, ρ(0) = ρ0, c(0) = c0 em Ω, (8)

onde QT = Ω × (0, T ), com Ω um domınio aberto e limitado de Rn, n = 2, 3. As incognitas sao: a funcao u

que representa o campo de velocidade do fluido, a pressao p, a densidade de carga livre ρ, o campo de fase c e o

potencial quımico ϕ. M e a mobilidade do campo de fase, D(u) := (∇u+∇uT )/2 e a parte simetrica do gradiente

e T : L2(Ω) → (H1(Ω))n e um operador linear satisfazendo

∥T (ρ)∥H1 ≤ C∥ρ∥L2 . (9)

O sistema (1)-(8) foi obtido por meio do modelo proposto por Yang, Li e Ding em [2]. Tomamos a densidade de

massa volumetrica ρ0, a constante dieletrica ϵγ e a condutividade dieletrica σ constantes positivas. Assumimos que

a mobilidade do campo de fase M depende de c. Admitimos a condicao de contorno de Neumann homogenea para

c, V e ∆c, ja para u colocamos a condicao de Dirichilet homogenea. Alem disso, supomos que a carga livre total

e nula, isso e,∫Ωρ = 0. Por fim, trocamos o operador T : L2(Ω) → (H1(Ω))n, n = 2, 3, dado por T (ρ) = −∇V ,

onde V e a solucao de ∆V = −ρ em Ω, ∂V∂n = 0 sobre ∂Ω, por um operador mais generico, satisfazendo apenas a

condicao . Depois desse procedimento, as constantes que apareceram nas equacoes, por simplicidade, foram tomadas

todas iguais a um.

As hipoteses assumidas sobre Ψ foram tais que o caso de um polinomio de quarta ordem com mınimos em 0

e 1, fosse contemplada. Este polinomio e conhecido como potencial de poc o duplo (double well potential) e e o

potencial que aparece no modelo original de Yang e Ding.

209

210


Os principais resultados sao os teoremas a seguir, cujas demonstracoes podem ser encontradas em [1], no capıtulo

4 .

Teorema 2.1. Suponha que Ω ⊂ Rn, n = 2, 3 e (u0, ρ0, c0) ∈ H ×L2(Ω) ×H1(Ω). Entao, existem u, ρ, c e ϕ tais

que

u ∈ L∞(0, T ;H) ∩ L2(0, T ;V ), ρ ∈ L∞(0, T ;L2(Ω)) ∩H1(0, T ;W 1,3(Ω)′),

c ∈ L∞(0, T ;H1(Ω)) ∩ L2(0, T ;H3(Ω)) ∩H1(0, T ;H1(Ω)′), ϕ ∈ L2(0, T ;H1(Ω)),

∂u

∂t∈ L2(0, T ;V ′) se n = 2 e

∂u

∂t∈ L4/3(0, T ;V ′) se n = 3.

e satisfazem as seguintes equacoes:⟨∂u

∂t, v

⟩+

∫Ω

µ(c)D(u) : D(v) + bu(u;u, v) =

∫Ω

ρT (ρ) · v +

∫Ω

ϕ∇c · v, (1)

∂ρ

∂t+ ∇ · (uρ) + ρ = 0 em D′(QT ), (2)⟨∂c

∂t, z

⟩+ b(u; c, z) +

∫Ω

M(c)∇ϕ · ∇z = 0, (3)

ϕ = Ψ′(c) − ∆c q.t.p. em QT , (4)

u(0) = u0, c(0) = c0 q.t.p. em Ω e ρ(0) = ρ0 em H1(Ω)′, (5)

para todo v ∈ V e z ∈ H1(Ω) e no sentido das distribuicoes em t. Na equacao (1), D(u) : D(v) := tr(D(u)TD(v)).

A solucao (u, ρ, c, ϕ) e chamada solucao fraca para o problema (1)-(8).

Teorema 2.2 (Regularidade). Seja Ω ⊂ Rn, n = 2, 3. Suponha que u0 ∈ V , ρ0 ∈ Lp(Ω), p ≥ 3, c0 ∈ H2(Ω) e∂c0∂n = 0 sobre ∂Ω. Entao, u, c e ρ dados pelo Teorema 2.1 satisfazem as seguintes regularidades:

u ∈ L∞(0, T ;V ) ∩ L2(0, T ;H2(Ω)) ∩H1(0, T ;L2(Ω)),

c ∈ L∞(0, T ;H2(Ω)) ∩ L2(0, T ;H4(Ω)) ∩H1(0, T ;L2(Ω)),

ρ ∈ L∞(0, T ;Lp(Ω)) ∩H1(0, T ;H1(Ω)′).

para todo T > 0 no caso em que Ω ⊂ R2 e para T = T∗, com T∗ suficientemente pequeno, para o caso em que

Ω ⊂ R3.

References

[1] pereira, a. f. Estudo de Boa Colocacao para Modelos Isotermicos de Campo de Fase Envolvendo Fluidos

Multifasicos. Tese (Doutorado em Matematica) - Instituto de Matematica, Estatıstica e Computacao Cientıfica,

Universidade Estadual de Campinas, Campinas - SP, 2019.

[2] yang, q., li, b.q. and ding, y. - 3D phase field modeling of electrohydrodynamic multiphase flows. Int. J.

Multiphas. Flow, 57, 1-9, 2013.


TWO-DIMENSIONAL INCOMPRESSIBLE MICROPOLAR FLUIDS MODEL WITH SINGULAR

INITIAL DATA

CLEYTON N. L. DE C. CUNHA1, ALEXIS BEJAR-LOPEZ2 & JUAN SOLER3

1Universidade Federal do Delta do Parnaıba, UFDPAR, PI, Brazil, [email protected],2Departamento de Matematica Aplicada and Research Unit “Modeling Nature” (MNat), Facultad de Ciencias, Universidad

de Granada, UGR, Spain, [email protected],3Departamento de Matematica Aplicada and Research Unit “Modeling Nature” (MNat), Facultad de Ciencias, Universidad

de Granada, UGR, Spain, [email protected]

Abstract

This work deals with Cauchy problem for the two-dimensional incompressible micropolar fluids model through

the velocity-vorticity formulation. It is assumed null angular viscosity and singular initial data, which includes

the possibility of vortex sheets or measures as initial data in Morrey spaces. By means of integral techniques we

establish the existence of weak solutions local and global in time. Also, the uniqueness and stability for these

solutions are analyzed.

1 Introduction

The incompressible micropolar fluid motion in R2, in the velocity-vorticity formulation, is described by the following

coupled system

∂tω − (ν + κ)∆ω + (u · ∇)ω = −2κ∆b+ curl f ,

∂tb− γ∆b+ 4κb+ (u · ∇)b = 2κω + g,

u = K ∗ ω, (1)

ω(·, 0) = ω0, b(·, 0) = b0,

where K is the Biot-Savart kernel, that is,

K(x) =1

2π|x|−2(−x2, x1), x ∈ R2. (2)

Here u is the velocity field, b is the microrotation field interpreted as the angular velocity field of rotation of

particles, ω = curl u is the vorticity field, f and g are given external fields, and the constants ν, κ, γ are viscosities

coefficients. We assume, without loss of generality, f = 0 and g = 0. The velocity fields given by (1) may include,

in particular, the case of vortex sheet.

In the Navier-Stokes case (κ = 0 and b = g = 0 in the micropolar model), there are several works with singular

initial data, for instance [2, 3, 5], and the references therein. In all these papers the parabolic character of the

vorticity equation was of great importance. In the micropolar case, this was also a key argument in [4], in the

particular case of null angular viscosity (γ = 0), to prove the global existence and uniqueness of smooth solutions

for the micropolar model with initial data in Hs(R2), s > 2. Motivated by these works, the goal of this paper is to

analyze the Cauchy problem associated with the two dimensional micropolar fluids with partial viscosity, namely

γ = 0, in terms of the evolution of the singular initial vorticity. We are also interested in the asymptotic behavior

of the micropolar fluid motion with respect to time t > 0 as well as in the case of vortex sheets structure of vorticity

described above.

211

212

2 Main Results

We use the standard notation for the Lebesgue and Sobolev spaces and we denote the Morrey-type space of measures

by Mp(Rn).

Theorem 2.1. Let ω0, b0 ∈ L1(R2) ∩ L∞(R2). Then, the system (1) has a unique global mild solution and the

inequality

∥u(·, t)∥∞ ≤ C(ν, κ)(ν + κ)t−1/2 (1)

holds, with t ∈ (0, T ], for all T > 0. Moreover, assume that (ω,u, b) and (ω, u, b) are mild solutions of system (1)

with initial data (ω0, b0) and (ω0, b0), respectively. Then, the following inequalities are verified

∥(u− u)(·, t)∥∞ ≤ Ct−1/2, (2)

∥(ω − ω)(·, t)∥1 + ∥(ω − ω)(·, t)∥∞ + ∥(b− b)(·, t)∥1 + ∥(b− b)(·, t)∥∞ ≤ CΠ, (3)

where Π = Π(ω0, b0, ω0, b0) = max∥ω0 − ω0∥1, ∥ω0 − ω0∥∞, ∥b0 − b0∥1, ∥b0 − b0∥∞ and C > 0 is a constant

independent of Π.

Theorem 2.2. Let ω0 ∈ M(R2) and b0 ∈ M(R2)∩Mp(R2), with p > 2. The system (1) has a unique global weak

solution such that

∥u(·, t)∥∞ ≤ C(ν, κ)(ν + κ)t−1/2,

with t ∈ (0, T ], for all T > 0. Moreover, the solutions are also stable in an analogous sense to (2)-(3), where in this

case Π depends on the norms M(R2) and Mp(R2) of the initial data.

Acknowledgements. This work has been partially supported by the MINECO-Feder (Spain) research grant

number RTI2018-098850-B-I00, the Junta de Andalucıa (Spain) Project PY18-RT-2422 & A-FQM-311-UGR18 and

by the CAPES/PRINT (Brazil) - Finance Code 001, #8881.311964/2018-01 (Cleyton Cunha) and the MECD

(Spain) research grant FPU19/01702 (Alexis Bejar-Lopez).

References

[1] bejar-lopez, a.; cunha, c.; soler, j. - Two-dimensional incompressible micropolar fluids model with

singular initial data, Preprint.

[2] benfatto, g.; esposito, r.; pulvirenti, m. - Planar Navier-Stokes flow for singular initial data, Nonlinear

Anal. 9, no. 6, 533-545, 1985.

[3] cottet, g-h. - Equations de Navier-Stokes dans le plan avec tourbillon initial mesure. (French) [Weak

solutions of 2-D Navier-Stokes equations with measure initial vorticity] C. R. Acad. Sci. Paris Ser. I Math.

303, no. 4, 105-108, 1986.

[4] dong, b-q.; zhang, z. - Global regularity of the 2D micropolar fluid flows with zero angular viscosity. (English

summary) J. Differential Equations 249, no. 1, 200-213, 2010.

[5] giga, y.; miyakawa, t.; osada, h. - Two-dimensional Navier-Stokes flow with measures as initial vorticity,

Arch. Rational Mech. Anal. 104, no. 3, 223-250, 1988.


ESTABILIZACAO NA FRONTEIRA NAO LINEAR DE UM SISTEMA TERMOELASTICO

EIJI RENAN TAKAHASHI1 & JUAN AMADEO SORIANO PALOMINO2

1PMA-Universidade Estadual de Maringa, UEM, PR, Brasil, [email protected] de Matematica da Universidade Estadual de Maringa, UEM, PR, Brasil, [email protected]

Resumo

Neste trabalho sera apresentado a existencia e estabilizacao da solucao de um sistema termoelastico com

dissipacao nao linear na fronteira. Provaremos inicialmente a existencia atraves da Teoria de Semigrupos de

Operadores Nao Lineares. Posteriormente para analise da estabilizacao utilizamos um metodo que consiste em

perturbar adequadamente a energia do sistema.

1 Introducaoo

Nosso objetivo e estudar o seguinte sistema termoelastico

utt − uxx + θx = 0 (1)

θt − θxx + uxt = 0 (2)

Para 0 < x < L e 0 < t < +∞, com as seguintes condicoes de fronteira.

u(0, t) = 0,

ux(L, t) = −g(ut(L, t)),

θ(0, t) = 0, θ(L, t) = 0,

(3)

e condicao inicial

u(x, 0) = u0(x), ut(x, 0) = u1(x). (4)

Onde g : R → R e uma funcao contınua decrescente que satisfaz,

∃λ > 0, c > 0; |g(s)| ≤ c|s|λ; |s| ≤ 1, (5)

∃c > 0; |g(s)| ≤ c|s|; |s| ≥ 1, (6)

∃p > 1, c > 0; g(s)s ≥ c|s|p+1; |s| ≤ 1, (7)

∃c > 0; g(s)s ≥ c|s|2; |s| ≥ 1. (8)

A energia do sistema (1)-(4) e dada por

E(t) =1

2

∫ L

0

u2tdx+1

2

∫ L

0

u2xdx+1

2

∫ L

0

θ2dx.

O Espaco de fase H e dado por

H = V × L2(0, L) × L2(0, L),

onde V =u ∈ H1(0, L); u(0) = 0

.

Para obter a estabilizacao fizemos como em [2].

213

214


Para mostrar a existencia de solucao do sistema (1)-(4), utilizamos a teoria encontrada em [1] e [3].

Nosso principal resultado e que para o sistema (1)-(4), utilizando as hipoteses para g acima citadas e com o

metodo da energia perturbada obtemos,

(i): Se λ = p = 1, existem constantes M > 1, γ > 0 tais que

E(t) ≤ME(0)e−γt ∀t ≥ 0.

(ii): Se λ > 1 e p > 1, existe uma constante M que depende de E(0) tal que

E(t) ≤ 4(Mt+ (E(0))

−(p−1)2

)−2/(p−1)

∀t ≥ 0.

(iii): Se λ < 1 e p > 1, existe uma constante M que depende de E(0) tal que

E(t) ≤ 4(Mt+ (E(0))

−p+1−2λ2λ

)−2λ/(p+1−2λ)

∀t ≥ 0.

Referencias

[1] Brezis, H. - Operateurs Maximaux Monotones et semi-groupes de contractions dans les espaces de Hilbert.,

North-Holland Mathematics Studies, N5. Notas de Matematica (50). North-Holland Publishing Co./American

Elsevier Publishing Co., Inc., Amsterdam- London/New York, 1973.

[2] Enrique Zuazua. - Controlabilidad Exata y Estabilizacion de La Equacion de Ondas 28040 Madrid, 1990.

[3] Nicolae H. Pavel. - Nonlinear Evolution Operators and Semigroups: Applications to Partial Differential

Equations (Lecture Notes in Mathematics, 1260. Springer, 1987.


AN INVERSE PROBLEM FOR A SIR REACTION-DIFFUSION MODEL

ANIBAL CORONEL1 & FERNANDO HUANCAS2

1Departamento de Ciencias Basicas, Universidad del Bıo-Bıo, Chillan, Chile, [email protected],2 Departamento de Matematicas, Universidad Tecnologica Metropolitana, Santiago, Chile, [email protected]

Abstract

In this work we study an inverse problem that arises in the problem of determining coefficients for a reaction-

diffusion system, originated in the theory of mathematical epidemiology. We consider a population lives in a

three-dimensional space is subdivided into the subclasses of susceptible, infected and recovered. We assume that

the dynamic process of disease transmission is governed by reaction-diffusion system. The inverse problem is

the identification of reaction coefficients. We apply the optimal control theory approach: the inverse problem

is reformulated as an optimization problem. Our results are the following: the existence and uniqueness of the

solution of the direct problem, the existence of solution for the adjoint system, the existence of the solution of

the optimization problem, a necessary optimality condition of first order, and the local uniqueness of the inverse

problem.

1 Introduction

In recent decades there is a growing interest in inverse problems that arise in mathematical models from various

applications and where the governing equations are given in terms of partial differential equations, see for example

[1, 2, 3, 3, 5, 6]. Particularly, we have the following SIR-type reaction-diffusion system

St − α∆S = µ(N − S) − βSI, It − α∆I = −(µ+ ν)I + βSI Rt − α∆R = νI − µR in QT , (1)

∇S · η = ∇I · η = ∇R · η = 0, on ∂Ω × [0, T ], (2)

(S, I,R)(x, 0) = (S0, I0, R0)(x), xin Ω, (3)

where S, I and R are the susceptible, infected and recovered densities of a population; α is the diffusion; and

β, µ, and ν are space dependent coefficients. The inverse problem is the identification of the reaction coefficients

from the final observation time of the state variables S, I and R: “Find the coefficients β, µ, ν such that at time

t = T the solution of system (1)-(3) is very close to the observed data Sobs, Iobs, and Robs ”. It can be reformulated

as the following optimization problem

inf J(S, I,R;β, µ, ν) : (β, µ, ν) ∈ Uad(Ω) y (S, I,R) is solution of (1)-(3), (4)

where

J =∥∥(S, I,R)(·, T ) − (Sobs, Iobs, Robs)

∥∥2L2(Ω)

+Γ

2∥∇(β, µ, ν)∥2L2(Ω), Uad(Ω) = A(Ω) ∩

[H [[d/2]]+1(Ω)

]3for d = 1, 2, 3, A(Ω) =

(β, µ, ν) ∈ [Cα(Ω)]3 : Ran(β, µ, ν) ⊂⊂]0, 1[3 ∇(β, µ, ν) ∈ [L2(Ω)]3

.

2 Main Results

We consider the hypotheses: (S0) The bounded and convex open set Ω is such that ∂Ω is C1; (S1) The initial

conditions S0; I0 and R0 are of class C2,α(Ω) and satisfy the inequalities

S0(x) ≥ 0, I0(x) ≥ 0, R0(x) ≥ 0;

∫Ω

I0(x)dx > 0,

∫Ω

R0(x)dx > 0, (S0 + I0 +R0) ≥ ϕ0 > 0,

215

216

on Ω, for some positive constant ϕ0; and (S2) the observation functions Sobs; Iobs y Robs are in L2(Ω). Then, the

main results of this work are the following.

Theorem 2.1. Suppose that hypotheses (S0)-(S2) are satisfied and further assume that (β, µ, ν, ) ∈ Cα(Ω) ×Cα(Ω) × Cα(Ω).. Then the direct problem ( (1)-(3)) admits a unique positive classical solution (S, I,R) such that

S, I,R ∈ C2+α,1+α/2(QT ) and also S; I y R are bounded on QT for any T ∈ R+.

Theorem 2.2. Suppose the hypotheses (S0)-(S2) hold. Then there is at least one solution to the optimization

problem.

Theorem 2.3. Suppose the hypotheses (S0)-(S2) hold.Consider that (β, µ, ν) is the solution of the inverse problem

and that (S, I,R) is the corresponding solution of the SIR model with (β, µ, ν) instead of (β, µ, ν). Then (p1, p2, p3)

is bounded L∞(0, t;H2(Ω)) for almost every time in ]0, T ]. In particular (p1, p2, p3) is bounded in L∞(0, t;L∞(Ω))

for almost all times in ]0, T ].

Theorem 2.4. Let (S, I,R) and (β, µ, ν) be as in theorem (2.3). Then∫Q

(µ− µ)[(N − S)p1 − Ip2 −Rp3] + SI(β − β)(p2 − p1) + I(ν − ν)(p3 − p2)

dxdt

+ C

∫Ω

|∇β∇(β − β)| + |∇µ∇(µ− µ)| + |∇ν∇(ν − ν)|

dx ≥ 0,

for any β, µ, ν ∈ Uad holds.

Theorem 2.5. Given c = (c1, c2, c3) ∈ R3+ (fixed) and defining Uc(Ω) =

(β, µ, ν) ∈ Uad(Ω) :

∫Ω

(β, µ, ν)dx = c.

Then there exists Γ ∈ R+ such that the solution of the inverse problem is only defined, except for an additive

constant, on Uc(Ω) in the sense L2(Ω) for any regularization parameter Γ > Γ.

References

[1] anger, g. - Inverse Problems in Differential Equations., Plenum Press, New York, 1990.

[2] coronel, a.. huancas, f, and sepulveda, m. - A note on the existence and stability of an inverse problem

for a SIS model. Computers and Mathematics with Applications, 77, 3186-3194, 2019.

[3] coronel, a.. huancas, f, and sepulveda, m. - On an inverse problem arising in an indirectly transmitted

diseases model. Inverse Problems, 35, 1-20, 2019.

[4] huili, x., and bin, l. - Solving the inverse problem of an SIS epidemic reaction-diffusion model by optimal

control methods. Compueters and Mathematical with Applications, 70, 805-819, 2015.

[5] marinova, t.t. marinova, r,s,omojola,j, and jackon, m. - Inverse problem for coefficient identification

in SIR epidemic models. Comput. Math. Appl. , 67, 2218-2227, 2014.

[6] sakthivel, k. gnanavel, s,balan, n,b, and balachandran, k. - Inverse problem for the reaction diffusion

system by optimization method. Appl. Math. Model. , 35, 571-579, 2011.


SOLVABILITY OF THE FRACTIONAL HYPERBOLIC KELLER-SEGEL SYSTEM

GERARDO HUAROTO1 & WLADIMIR NEVES2

1Instituto de Matematica, UFAL, AL, Brasil, [email protected],2Instituto de Matematica, UFRJ, RJ, Brasil, [email protected]

Abstract

We study a new nonlocal approach to the mathematical modelling of the Chemotaxis problem, which describes

the random motion of a certain population due a substance concentration. Considering the initial- boundary

value problem for the fractional hyperbolic Keller-Segel model, we prove the solvability of the problem. The

solvability result relies mostly on the kinetic method.

1 Introduction

We introduce and study in this paper the Fractional Hyperbolic Keller-Segel (FHKS for short) model for chemotaxis

described by the following system

∂tu+ div(g(u)∇Kc

)= 0, in (0,∞) × Ω,

(−∆N )1−s c+ c = u, in Ω,

u|t=0 = u0, in Ω,

∇Kc · ν = 0, on Γ,

(1)

where u(t, x) is the density of cells and c(t, x) is the chemoattractant concentration, which is responsible for the

cell aggregation. The problem is posed in a bounded open subset Ω ⊂ Rn, (n = 1, 2, or 3), with C2−boundary

denoted by Γ, and as usual we denote by ν(r) the outward normal to Ω at r ∈ Γ. The given measurable bounded

function u0 is the initial condition of the cells, and we assume

0 ≤ u0(x) ≤ 1, for a.e. x ∈ Ω. (2)

Moreover, since the normal flux in the equation on u vanishes on Γ, that is the boundary is characteristic, it is

not necessary to prescribe boundary conditions for u, which prevents some specific difficulties related to the trace

problem, see [5] for instance. Here for 0 < s < 1, (−∆N )s denotes the Neumann spectral fractional Laplacian

(NSFL for short) operator, which characterizes long-range diffusion effects. We also consider the non-local operator

K ≡ (−∆N )−s.

The theory of chemotaxis modeling goes back to E. F. Keller and L. A. Segel [2, 3, 4], where a detailed description

of the movement of cells oriented by chemical cues can be found. In fact, a nonlocal version of the Keler-Segel

model has been proposed by Caffarelli, Vazquez in [1]. Although, the fractional model proposed here in (1) is a

closer (fractional) generalization of the model considered in Dalibard, Perthame [6]. Indeed, in that paper they

studied the following system

∂tu+ div(g(u)∇S

)= 0, in (0,∞) × Ω,

(−∆)S + S = u, in Ω,

u|t=0 = u0, in Ω,

∇S · ν = 0, on Γ,

(3)

217

218

which follows from the system (1), at least formally passing to the limit as s→ 0+.

2 Main Results

We begin observing that, the first equation in (1) is a hyperbolic scalar conservation law, thus the density of cells

function u may admit shocks. Therefore, in order to select the more correct physical solution, we need an admissible

criteria, which is given by the entropy condition.

Now, we are able to state plainly the main result of this paper. Then, we have the following

Theorem 2.1 (Main Theorem). Let u0 ∈ L∞(Ω) be an initial data satisfying (2) and s ∈ (0, 1). Then, there exists

a pair of functions

(u, c) ∈ L∞((0,∞) × Ω) × L∞((0,∞);D((−∆N )1−s)),

which is a weak entropy solution to the FHKS system, and it satisfies

0 ≤ u(t, x) ≤ 1, 0 ≤ c(t, x) ≤ 1,

for almost all t > 0 and x ∈ Ω.

References

[1] Caffarelli, L.; Vazquez, J.L. Nonlinear porous medium flow with fractional potential pressure. Arch.

Ration. Mech. Anal. 202, (2011), no. 2, 537–565.

[2] Keller, E. F., Segel, L. A., Initiation of slime mold aggregation viewed as an instability J. Theor. Biol.

26, (1970), 339–415.

[3] Keller, E. F., Segel, L. A., Model for chemotaxis. J. Theor. Biol. 30, (1971), 225–234.

[4] Keller, E. F., Segel, L. A., Travelling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol.

30, (1971), 235–248.

[5] Neves, W., Panov, E., Silva, J., Strong traces for conservation laws with general nonautonomous flux,

SIAM J. Math. Anal., 50 6 (2018), 6049–6081.

[6] Perthame, B.; Dalibard, A.–L. Existence of solutions of the hyperbolic Keller–Segel model. Trans. Amer.

Math. Soc. 361, (2009), no. 5, 2319–2335.


CONTROLLABILITY OF PHASE-FIELD SYSTEM WITH ONE CONTROL

SOUSA-NETO, G. R.1 & GONZALEZ-BURGOS, M.2

1Departamenteo de Matematica, UFPI, PI, Brasil, [email protected],2Dpto. Ecuaciones Diferenciales y Analisis Numerico and IMUS, Facultad de Matematicas, Universidad de Sevilla, Espanha

Abstract

In this paper, we present some controllability results for linear and nonlinear phase-field systems of Caginalp

type considered in a bounded interval of Rwhen the scalar control force acts on the temperature equation of the

system by means of the Dirichlet condition on one of the endpoints of the interval. In order to prove the linear

result we use the moment method providing an estimate of the cost of fast controls. Using this estimate we

prove a local exact boundary controllability result to constant trajectories of the nonlinear phase-field system.

2020 Elsevier Inc. All rights reserved.

1 Introduction

In this work, we present a controllability results for a nonlinear phase-field systems of Caginalp type considered in

a bounded interval of R when the scalar control force acts on the temperature equation of the system by means of

the Dirichlet condition on one of the endpoints of the interval. We use the moment method providing an estimate of

the cost to achieve a local exact boundary controllability result to constant trajectories of the nonlinear phase-field

system.

The Phase-field system it is a model describing the transition between the solid and liquid phases in

solidification/melting processes of a material occupying a domain. Given a time T > 0 and the cylinder

QT := (0, π) × (0, T ), the system is described as follows by G. Caginalp in [1]:

θt − ξθxx +1

2ρξϕxx +

ρ

τθ = f(ϕ) in QT ,

ϕt − ξϕxx − 2

τθ = −2

ρf(ϕ) in QT ,

θ(0, ·) = v, ϕ(0, ·) = c, θ(π, ·) = 0, ϕ(π, ·) = c on (0, T ),

θ(·, 0) = θ0, ϕ(·, 0) = ϕ0 in (0, π),

In the Phase-field system above θ = θ(x, t) is the temperature of the material and ϕ = ϕ(x, t) identifies the

phase transition of the material. When ϕ = 1 the material is in the solid state, and when ϕ = −1 it means that

the material is in the liquid state.

Also, θ0, ϕ0 represents the initial data; v ∈ L2(0, T ) is the control; c is a constant assuming the possible values

in the set −1, 0, 1; the constants ρ, τ, ξ are, respectively, the latent heat, the relaxation time, and the thermal

diffusivity; and f(ϕ) is the nonlinear part of the system given by

f(ϕ) = − ρ

4τ

(ϕ− ϕ3

).

2 Main Results

The main result of the work is given in the following.

219

220

Theorem 2.1. Let us fix T > 0 and assume that ξ2τ2(ℓ2 − k2)2 − 2ξρτ(ℓ2 + k2) − 2ρ− 1 = 0, ∀k, ℓ ≥ 1, ℓ > k,

ξ = 1

j2ρ

τ, ∀j ≥ 1.

Then, there exists ϵ > 0 such that, for any (θ0, ϕ0) ∈ H−1 × (c+H10 ), with ∥θ0∥H−1 + ∥ϕ0 − c∥H1

0≤ ϵ, there exists

v ∈ L2(0, T ) for which system (1) has a unique solution which satisfies θ(·, T ), ϕ(·, T ) = (0, c) in (0, T ).

Proof The proof is developed using the following strategy.

First we prove the null controllability at time T > 0 of the homogeneous linearized system (after a change of

variables (θ0, ϕ0, θ, ϕ) = (θ0, ϕ0 − c, θ, ϕ− c))

θt − ξθxx +1

2ρξϕxx − ρ

2τϕ+

ρ

τθ = 0 in QT ,

ϕt − ξϕxx +1

τϕ− 2

τθ = 0 in QT ,

θ(0, ·) = v, ϕ(0, ·) = θ(π, ·) = ϕ(π, ·) = 0 on (0, T ),

θ(·, 0) = θ0, ϕ(·, 0) = ϕ0 in (0, π),

The assumptions on ξ, ρ, τ are used in the Moment Method for assuring that the eigenvalues of the space operator

of the homogeneous linear system satisfy suitable properties to produce the following estimate of the control cost:

∥v∥L2(0,T ) ≤ C0 eMT ∥y0∥H−1 .

Next, we prove the null controllability at time T > 0 of the non-homogeneous linearized system

θt − ξθxx +1

2ρξϕxx − ρ

2τϕ+

ρ

τθ = f1 in QT ,

ϕt − ξϕxx +1

τϕ− 2

τθ = f2 in QT ,

θ(0, ·) = v, ϕ(0, ·) = θ(π, ·) = ϕ(π, ·) = 0 on (0, T ),

θ(·, 0) = θ0, ϕ(·, 0) = ϕ0 in (0, π),

where f = (f1, f2) is a source with exponential decay when t→ T :

eC

T−t f ∈ L2(QT ), for suitable C > 0.

Finally, we apply a Fixed-Point argument to the non-homogeneous linear system with the operator

(f1, f2) 7→(±3ρ

4τϕ2 +

ρ

4τϕ3,∓3ρ

2τϕ2 − 1

2τϕ3)

in order to recover the local null controllability result at time T for the nonlinear Phase-Field system.

References

[1] Caginalp, G. - An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal. 92 (1986),

no. 3, 205–245.

[2] Liu, Takahashi, Tucsnak - ESAIM Control Optim. Calc. Var. 19 (2013), no. 1, 20–42.

[3] H.O. Fattorini, D.L. Russel - Exact controllability theorems for linear parabolic equations in one space

dimension, Arch. Ration. Mech. Anal. 43 (1971) 272-292.


CONTROLABILIDADE EXATA PARA A EQUACAO KDV VIA ESTRATEGIA

STACKELBERG-NASH

ISLANITA C. A. ALBUQUERQUE1, FAGNER D. ARARUNA2 & MAURICIO C. SANTOS3

1Campus Mata Norte, UPE, PE, Brasil, [email protected],2Departamento de Matematica, UFPB, PB, Brasil, [email protected],

3Departamento de Matematica, UFPB, PB, Brasil, [email protected]

Abstract

Neste trabalho abordamos um problema de controle hierarquico para a equacao em Korteweg-de Vries (KdV)

com controles distribuıdos seguindo uma estrategia de Stackelberg-Nash. Tratamos de um problema de controle

onde muitos objetivos devem ser alcancados de uma so vez, contamos com um controle principal chamado lıder,

e dois controles secundarios chamados seguidores, onde cada um deles tem seu proprio objetivo. O objetivo do

lıder e conduzir as solucoes da equacao KdV para uma determinada trajetoria, enquanto os seguidores devem

ter um equilıbrio de Nash para seus objetivos.

1 Introducao

A equacao de Korteweg-de Vries (KdV) e uma equacao diferencial parcial de terceira ordem que modela a propagacao

das ondas em superfıcies de aguas rasas.

Neste trabalho consideramos um problema de controle multi-objetivo (isto e, muitos objetivos devem ser

cumpridos de uma vez) o que pode tornar sua solucao inviavel. Para superar isto, aplicamos o conceito de Otimizacao

de Stackelberg onde uma hierarquia para os controles e assumida. Consideramos um controle denominado lıder e

outros controles chamados de seguidores. Uma vez que a escolha do lıder e fixada, os seguidores devem cumprir

seus objetivos de forma otimizada.

Vamos ser mais especıficos. Seja (0, L) ⊂ R um intervalo aberto e T > 0 um numero real. Nos consideramos

controles internos suportados em um subconjunto aberto nao vazio ω ⊂ (0, L) e condicoes homogeneas de fronteiras.

Definimos Q = (0, L) × (0, T ) e para algum subconjunto aberto ω ⊂ (0, L) definimos Qω = ω × (0, T ).

Consideremos a equacao KdV nao linearyt + yx + yxxx + yyx = f1O + v1χO1 + v2χO2 in Q,

y (0, ·) = y (L, ·) = yx (L, ·) = 0 in (0, T ) ,

y (x, ·) = y0 in (0, L) ,

(1)

onde y = y(x, t) e o estado e y0 e dado. Em (1), o conjunto O ⊂ (0, L) e o domınio do controle lıder f e

O1,O2 ⊂ (0, L) sao os domınios dos controles seguidores v1 e v2 (todos supostos bem pequenos e disjuntos). A

funcao 1A representa a funcao caracterıstica de um conjunto aberto A, onde χA e uma funcao C∞0 (A).

Sejam O1,d, O2,d ⊂ (0, L) conjuntos abertos e considere os funcionais

Ji(y0, f, v1, v2) =

αi

2

∫∫Q

χOi,d|y − yi d|2 dxdt+

µi

2

∫∫Q

χOi|vi|2 dxdt, i = 1, 2, (2)

onde αi > 0, µi > 0 sao constantes e yi,d = yi,d(x, t) sao funcoes dadas.

A controlabilidade exata de Stackelberg-Nash para equacao KdV pode ser descrita em duas etapas. A primeira,

para f fixado, os seguidores v1 e v2 buscam ser um equilıbrio de Nash para os funcionais custos Ji (i = 1, 2). (Isto

e, procuramos pelo par (v1, v2) com vi ∈ L2(Oi × (0, T )) tal que satisfaca Ji(f ; v1, v2) = minvi

Ji(vi), i = 1, 2).

221

222

Para a segunda etapa, fixamos uma trajetoria y, que e solucao suficientemente regular de um sistema, sob

mesmas condicoes de fronteira e dado inicial de (1).

Uma vez que o equilıbro de Nash foi encontrado e para cada f fixado, procuramos por um controle f ∈L2(O × (0, T )) tal que y(·, T ) = y(·, T ), isto e, satisfaz a condicao de controlabilidade exata em (0, L).

Entao definimos a nova variavel z = y − y e zi,d = yi,d − y, e mostramos a controlabilidade nula para z que e

z(x, T ) = 0 em (0, L) (Observe que mostrar isto e equivalente a mostrar a controlabilidade exata para y). Onde z

junto com ϕi (i = 1, 2) satisfazem um sistema de otimalidade do tipo:

zt + zx + zxxx + zzx + (yz)x = f 1O −2∑

i=1

1

µiϕiχOi

+ f0 em Q,

−ϕit − ϕix − ϕixxx − (z + y)ϕix = αi(z − zi,d)χOi,d+ f i, em Q

z (0, ·) = z (L, ·) = zx (L, ·) = 0 em (0, T ) ,

ϕi (0, ·) = ϕi (L, ·) = ϕix (0, ·) = 0 em (0, T ) ,

z(·, 0) = z0, ϕi(·, T ) = 0 em (0, L) .

(3)

Deste modo, a estrategia que adotamos para encontrar o equilıbrio de Nash consiste em provar que o sistema (3)

possui solucoes. Uma vez o equilıbrio de Nash encontrado temos que provar que e possıvel resolver simultaneamente

o objetivo do lıder, ou seja, temos que provar a existencia de f tal que a solucao de (3) satisfaca a condicao de

controlabilidade nula de z, o que motiva o resultado principal deste trabalho.


Teorema 2.1. Para i = 1, 2, suponha que

Oi,d ∩ O = ∅ (1)

e que µi sao suficientemente grandes. Alem disso, suponha que uma das duas condicoes e satisfeita:

O1,d = O2,d ou O1,d ∩ O = O2,d ∩ O. (2)

Entao, existe uma funcao positiva ρ = ρ(t) explodindo em t = T e δ > 0 tal que se

∥z0∥2H10 (0,L) +

2∑i=1

∫∫Oi,d×(0,T )

ρ2|zi,d|2 dx dt < δ, (3)

existem controles f ∈ L2(O × (0, T )) tal que a solucao (y, ϕ1, ϕ2) de (3) satisfaz y(·, T ) = 0.

Prova: Existencia - Para mostrar o resultado, primeiro mostramos um resultado de controlabilidade nula para

um sistema linearizado do sistema (3). A prova do caso linear e feita atraves do Metodo de Unicidade de Hilbert

(HUM), que consiste em uma equivalencia a uma estimativa de observabilidade adequada para as solucoes de um

sistema adjunto, nesta etapa, novas estimativas de Carleman sao demonstradas e para elas sao construıdas novas

funcoes pesos pela necessidade de detalhes tecnicos [1]. Por fim, para o caso nao linear, utilizamos o Teorema da

Funcao Inversa.

References

[1] Araruna, F.D. - Cerpa, E., Mercado, A. and Santos, M.C. Internal null controllability of a linear

Schrodinger-KdV system on a bounded interval. , ScienceDirect, J. Differential Equations 260 (2016) 653-687.


EXPONENTIAL ATTRACTOR FOR A CLASS OF NON LOCAL EVOLUTION EQUATIONS

JANDEILSON S. DA SILVA1, SEVERINO H. DA SILVA2 & ALDO T. LOUREDO3

1UAMat, UFCG, PB, Brasil, [email protected],2UAMat, UFCG, PB, Brasil, [email protected],

3DM, UEPB, PB, Brasil, [email protected]

Abstract

In this work we study the existence of exponential attractor for a non local evolution equation, which

generalizes the model that describe the neuronal activity. Our results extend results obtained in [1] and in [7],

where the studied models are configured as particular cases of the model presented here. Furthermore, from the

existence of the exponential attractor, we conclude that the fractal dimension of the global attractor given in [2]

has finite fractal dimension.

1 Introduction

We consider the non local evolution problem∂tu(x, t) = −u(x, t) + g(βK(f u)(x, t) + βh), x ∈ Ω, t ∈ [0,∞[;

u(x, t) = 0, x ∈ RN \ Ω, t ∈ [0,∞[;

u(x, 0) = u0(x), x ∈ RN ,

(1)

where u(x, t) is a real function on RN × [0,∞[, Ω is a bounded smooth domain in RN (N ≥ 1); h and β are

nonnegative constants; K is an integral operator with symmetric kernel, given by Kv(x) :=∫RN J(x, y)v(y)dy

where J is a non negative symmetric function of class C1, with∫RN J(x, y)dy =

∫RN J(x, y)dx = 1. The functions f

and g f, g : R → R are locally Lipschitz continuous functions satisfying the growth conditions

|g(x)| ≤ k1|x| + k2 and |f(x)| ≤ c1|x| + c2, ∀ x ∈ R

for some non-negative constants k1, k2, c1, c2.

In addition, we will also assume that g ∈ C1(R) with

|g′(x)| ≤ k3|x| + k4, ∀ x ∈ R,

for some non-negative constants k3, k4 > 0; that g′ is Lipschitz on bounded sets and that function f satisfy

|f(x) − f(y)| ≤ c0(1 + |x|p−1 + |y|p−1)|x− y|, ∀ (x, y) ∈ R× R,

where c0 is some non-negative constant.

2 Main Results

In [2] it has been proven that, under the hypotheses mentioned, the problem (1) is well posed in the phase space

X = u ∈ Lp(Rn) : u(x) = 0, if x ∈ Rn\Ω, with the induced norm of Lp(Ω), which is isometric to Lp(Ω).

Furthermore, in [2] it has also been proven that the problem (1) generates a C1 flow, S(t)t≥0, in Lp(Ω).

We prove that space W 1,p(Ω) is positively invariant by the action of S(t) and admits a bounded subset B1,

wich is positively invariant and absorbing for S(t) (in the topology of W 1,p(Ω)). Using techniques of [4], [5] and

[6] we obtain the main results of this work.

223

224

Theorem 2.1. The flow S(t)t≥0 has a compact set M ⊂ Lp(Ω) with the following properties:

(i) The set M is positively invariant under semigroup S(t), that is, S(t)M ⊂ M for any t ≥ 0.

(ii) The set M has finite fractal dimension, that is, dimF (M, Lp(Ω)) <∞.

(iii) The set M attracts exponentially B1, that is, there exist α ≥ 0 and ω > 0 such that, for any t ≥ 0,

distLp(Ω)

(S(t)B1,M

)< αe−ωt,

where distLp(Ω) denotes the Hausdorff semidistance in Lp(Ω).

Remark 2.1. The set M given in the Theorem 2.1, attracts exponentially the action of S(t) on bounded subsets

of B1. This is a sense of expoenential attractor discussed in [3].

If, in addition to the hypotheses initially considered, we assume that f and g are Lipschitzian, we obtain the

following result, which gives an exponential attractor for S(t), in the sense defined in [7] and in [6], which attracts

exponentially any bounded subset of Lp(Ω).

Theorem 2.2. The set M given in the Theorem 2.1 is an exponential attractor for the semigroup S(t) generated

by the solutions of the equation (1).

Proof Due to the hypothese that f and g are Lipschitizian we have

distLp(Ω) (S(t)B,B1) ≤ C1(B)e[LgβLf∥J∥1−1]t.

Hence, using Theorem 2.1 and a result on transitivity of exponential attraction given in [5], the result follows.

Corollary 2.1. The global attractor A for the semigroup S(t) given in [2] has a finite fractal dimension.

References

[1] Da Silva, S.H.; Bezerra, F.D.M...Finite fractal dimensionality of attractors for non local evolution equations.

Electronic Journal of Differential Equations, 2013, No. 221, 1-9, 2013.

[2] Da Silva, S.H.; Garcia, A.R.G.; Lucena, B.E.P..Dissipative property for non local evolution equations. Bull.

Belg. Math. Soc. Simon Stevin. 26, 91-117, 2019.

[3] Eden, A., Foias, C., Kalantarov, V.. A Remark on Two Constructions of Exponential Attractors for α-

Contractions. Journal of Dynamics and Differential Equations, Vol. 10, No. 1, 37-45, 1998.

[4] Efendiev, M., Miranville, A., Zelik, S.. Exponential attractors for a nonlinear reaction-diffusion systems in R3.

C. R. Acad. Sci. Paris SA¨r. I Math. 330(8), 713-718, 2000.

[5] Fabrie, P., Galusinski, C., Miranville, A., Zelik, S. Uniform exponential attractors for a singularly perturbed

damped wave equation. Discrete and Continuous Dynamical Systems 10(2), 211-238, 2004.

[6] Milani, A.J., Koksch, N.J. An Introduction to Semiflows. Monographs and Surveys in Pure and Applied

Mathematics, vol 134. Chapman & Hall/CRC, Boca Raton, 2005.

[7] Shomberg, J.L.,Existence of global attractors and gradient property for a class of non local evolution equations,

Differ. Equ. Dyn Syst. 23, no.1, 99-115, 2015.


GLOBAL SOLUTIONS TO THE NON-LOCAL NAVIER-STOKES EQUATIONS

JOELMA AZEVEDO1, JUAN CARLOS POZO2 & ARLUCIO VIANA3

1Universidade de Pernambuco, UPE, PE, Brasil, [email protected],2Facultad de Ciencias, Universidad de Chile, Santiago, Chile, [email protected],

3Universidade Federal de Sergipe, UFS, SE, Brasil, [email protected]

Abstract

We study the global well-posedness for a non-local-in-time Navier-Stokes equation. Our results recover in

particular other existing well-posedness results for the Navier-Stokes equations and their time-fractional version.

We show the appropriate manner to apply Kato’s strategy and this context, with initial conditions in the

divergence-free Lebesgue space Lσd (Rd).

1 Introduction

Consider the fractional-in-time Navier-Stokes equation

∂αt u− ∆u+ (u · ∇)u+ ∇p = f, t > 0, x ∈ Ω ⊂ Rd,

∇ · u = 0, t > 0, x ∈ Ω ⊂ Rd,

u(0, x) = u0(x), x ∈ Ω ⊂ Rd,

where ∂αt u denotes the fractional derivative of u in the Caputo’s sense with order α ∈ (0, 1). If the product

(k ∗ v) denotes the convolution on the positive halfline R+ := [0,∞) with respect to time variable, then we have

∂αt u = g1−α ∗ ut, for an absolutely continuous function u, where gβ is the standard notation for the function

gβ(t) = tβ−1

Γ(β) , t > 0, β > 0. Toward the possibility of considering more general nonlocal-in-time effects, we will

replace gα by k, and we assume as a general hypothesis that k is a kernel of type (PC), by which we mean that the

following condition is satisfied:

(PC) k ∈ L1,loc(R+) is nonnegative and nonincreasing, and there exists a kernel ℓ ∈ L1,loc(R+) such that k ∗ ℓ = 1

on (0,∞).

We also write (k, ℓ) ∈ PC. We point out that the kernels of type (PC) are called Sonine kernels and they have been

successfully used to study integral equations of first kind in the spaces of Holder continuous, Lebesgue and Sobolev

functions, see [1].

Therefore, we consider the following problem for the following nonlocal-in-time Navier-Stoke-type equation

∂t(k ∗ (u− u0)) − ∆u+ (u · ∇)u+ ∇p = f, t > 0, x ∈ Rd, (1)

∇ · u = 0, t > 0, x ∈ Rd, (2)

u(0, x) = u0(x), x ∈ Rd, (3)

where u(t, x) represents the velocity field and p(t, x) is the associated pressure of the fluid. The function

u0(x) = u(0, x) is the initial velocity and f(t, x) represents an external force. The problem (1)-(3), can be written

in an abstract form as

∂t(k ∗ (u− u0)) + Apu = F (u, u) + Pf, t > 0, (4)

225

226

where Apu := P (−∆)u, P : Lp(Rd) → Lσp (Rd) is well-known as Helmholtz-Leray’s projection, and the nonlinear

term F (u, v) := −P (u · ∇)v. Equation (4) can be written as a Volterra equation of the form

u+ (ℓ ∗ Aru)(t) = u0 + (ℓ ∗ [F (u, u) + Pf ])(t), t > 0, (5)

by condition (k, ℓ) ∈ PC.

2 Main Results

We investigate the existence and uniqueness of global mild solutions for equation (5). Before we state the

main result, we introduce space where the mild solution will dwell. Let d ∈ N. For any 2 ≤ d < q < ∞,

consider the space Xq of the functions v satisfying v ∈ Cb([0,∞);Lσd (Rd)), (1 ∗ ℓ)

12−

d2q v ∈ Cb((0,∞);Lσ

q (Rd)) and

(1 ∗ ℓ) 12∇v ∈ Cb((0,∞);Lσ

d (Rd)), which is a Banach space with norm

∥v∥Xq:= maxsup

t>0∥v(t)∥Lσ

d (Rd), supt>0

[(1 ∗ l)(t)]12−

d2q ∥v(t)∥Lσ

q (Rd), supt>0

[(1 ∗ l)(t)] 12 ∥∇v(t)∥Lσ

d (Rd).

The existence of the mild solutions solution for (5) will be a consequence of the following fixed point lemma (see

[2, Lemma 1.5]).

Lemma 2.1. Let X be an abstract Banach Space and L : X×X → X a bilinear operator. Assume that there exists

η > 0 such that , given x1, x2 ∈ X, we have ∥L(x1, x2)∥ ≤ η∥x1∥∥x2∥. Then for any y ∈ X, such that 4η∥y∥ < 1,

the equation x = y + L(x, x) has a solution x in X. Moreover, this solution x is the only one such that

∥x∥ ≤1 −

√1 − 4η∥y∥2η

. (1)

Theorem 2.1. Let d ∈ N, 2 ≤ d < q < ∞, η an appropriate constant and f ∈ Cb([0,∞);L qdq+d

(Rd)) be such

that α := supt>0[(1 ∗ ℓ)(t)]1−d2q ∥f(t)∥L qd

q+d

(Rd) < ∞. For u0 ∈ Lσd (Rd) and α > 0 sufficiently small, there exists

0 < λ < 1−4αϑCη4η , where ϑ and C are positive real constants, such that if ∥u0∥Ld(Rd) ≤ min1, C−1λ, then the

problem (5) has a global mild solution u ∈ Xq that is the unique one satisfying (1). In particular,

∥u(t, ·)∥Lq(Rd) ≤1−

√1− 4η(λ+ αϑC)

2η[(1 ∗ ℓ)(t)]−

12+ d

2q and ∥∇u(t, ·)∥Ld(Rd) ≤1−

√1− 4η(λ+ αϑC)

2η[(1 ∗ ℓ)(t)]−

12 .

If, in addition, f ≡ 0, we have

[(1 ∗ ℓ)(t)]12−

d2q ∥u(t, ·)∥Lq(Rd) → 0 and [(1 ∗ ℓ)(t)] 1

2 ∥∇u(t, ·)∥Ld(Rd) → 0,

as t→ 0+. Furthermore, let u, v ∈ Xq be two solutions given by the existence part corresponding to the initial data

u0 and v0, respectively. Then,

∥u− v∥Xq≤ C√

1 − 4η(λ+ αϑC)∥u0 − v0∥Ld(Rd).

References

[1] carlone, r. and fiorenza, a. and tentarelli, l. - The action of Volterra integral operators with highly

singular kernels on Holder continuous, Lebesgue and Sobolev functions, J. Funct. Anal., 273, 1258-1294, 2017.

[2] cannone, m. - A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana,

13, 515-541, 1997.


CONTROLABILIDADE GLOBAL DO SISTEMA DE BOUSSINESQ COM CONDICOES DE

FRONTEIRA DO TIPO NAVIER

F. W. CHAVES-SILVA1, E. FERNANDEZ-CARA2, K. LE BALC’H3, J. L. F. MACHADO4 & D. A. SOUZA5

1Departamento de Matematica, UFPB, PB, Brasil, [email protected],2Universidad de Sevilla, EDAN e IMUS, Sevilla, Espanha, [email protected],

3Institut de Mathematiques de Bordeaux, Bordeaux, Franca, [email protected],4Instituto Federal do Ceara, IFCE, CE, Brasil, [email protected],5Universidad de Sevilla, EDAN e IMUS, Sevilla, Espanha, [email protected]

Abstract

Neste trabalho, apresentamos um resultado global de controlabilidade exata as trajetorias do sistema de

Boussinesq. Consideraremos domınios limitados com fronteiras suaves. Completaremos o modelo considerando

uma condicao de fronteira do tipo Navier slip-with-friction para o campo velocidade e uma condicao de fronteira

do tipo Robin e imposta a temperatura. Assumiremos que se pode atuar livremente sobre a velocidade e a

temperatura em uma parte arbitraria da fronteira. A prova se baseia em tres argumentos principais. Primeiro,

reformularemos o problema como um problema de controlabilidade distribuıda usando um procedimento de

extensao de domınio. Entao, provaremos um resultado global de controlabilidade aproximado seguindo a

estrategia de Coron et al [J. EUR. Math. Soc., 22 (2020), pp. 1625-1673], que trata das equacoes de Navier-

Stokes (o argumento depende da controlabilidade do sistema invıscido de Boussinesq e das expansoes assintoticas

do boundary layer). Finalmente, concluiremos com um resultado de controlabilidade local que estabeleceremos

por meio de um argumento de linearizacao e estimativas de Carleman apropriadas.

1 Introducao

Seja T > 0, Ω ⊂ Rn (n = 2, 3) um domınio limitado regular com Γ := ∂Ω e Γc ⊂ Γ um subconjunto aberto nao-

vazio que intercepta todas as componentes conexas de Γ.Consideraremos um sistema de Boussinesq, tal que sobre

a fronteira, o campo velocidade do fluido deve satisfazer uma condicao Navier slip-with-friction e a temperatura

uma condicao do tipo Robin. Assumimos tambem que o controle pode atuar em Γc, obtendo:

ut − ∆u+ (u · ∇)u+ ∇p = θen em (0, T ) × Ω,

θt − ∆θ + u · ∇θ = 0 em (0, T ) × Ω,

∇ · u = 0 em (0, T ) × Ω,

u · ν = 0, N(u) = 0 sobre (0, T ) × (Γ \ Γc) ,

R(θ) = 0 sobre (0, T ) × (Γ \ Γc) ,

u(0, · ) = u0, θ(0, · ) = θ0 em Ω.

(1)

As funcoes u = u(t, x), θ = θ(t, x) e p = p(t, x) sao, respectivamente, vistas como o campo de velocidade, a

temperatura e a pressao do fluido. Os termos das condicoes de fronteira de Navier e Robin sao, respectivamente,

dados pelas seguintes formulas:

N(u) := [D(u)ν +Mu]tan e R(θ) :=∂θ

∂ν+mθ,

onde M = M(t, x) e uma matriz simetrica regular relacionada a rugosidade da fronteira, chamada matriz de friccao

e m = m(t, x) e uma funcao regular, conhecida como coeficiente de transferencia de calor. Com estas condicoes

227

228

temos a presenca de boundary layer, devido o atrito na fronteira. Provamos que o sistema (1) e controlavel a

trajetorias, isto significa ser possıvel conduzir (por meio de controles) qualquer estado inicial a qualquer trajetoria

prescrita do sistema.


Vamos definir

L2c(Ω)n := u ∈ L2(Ω)n : ∇ · u = 0 in Ω, u · ν = 0 on Γ \ Γc,

WT (Ω) := [C0w([0, T ];L2

c(Ω)n) ∩ L2(0, T ;H1(Ω)n)] × [C0w([0, T ];L2(Ω)) ∩ L2(0, T ;H1(Ω))].

Temos o seguinte resultado principal:

Teorema 2.1. Sejam T > 0 um tempo positivo, (u0, θ0) ∈ L2c(Ω)n × L2(Ω) um dado inicial e (u, θ) ∈WT (Ω) uma

trajetoria fraca de (1). Entao, existe uma solucao fraca controlada para (1) em WT (Ω) que satisfaz

(u, θ) (T, · ) =(u, θ)

(T, · ).

Este Teorema generaliza para o sistema de Boussinesq (onde os efeitos termicos sao considerados) o principal

resultado do controle em [1], estabelecido para as equacoes de Navier-Stokes.

Esquema da prova: Principais ideias e resultados necessarios para a prova do Teorema:

Reduzimos o problema de controlabilidade distribuıda aplicando uma tecnica classica de extensao de domınio.

Em seguida, limitamos nossas consideracoes em dados iniciais regulares, usando o efeito de regularizacao do

sistema Boussinesq nao controlado.

Partindo de dados iniciais suficientemente regulares, provamos um resultado global de controlabilidade

aproximada. O boundary layer e tratado nesta etapa. Para isso, seguimos a estrategia realizada por Coron,

Marbach e Sueur em [1] no caso Navier-Stokes.

Provamos um resultado de controlabilidade local usando desigualdades de Carleman para o adjunto do sistema

linearizado e uma estrategia de ponto fixo. Para isso, utilizamos ideias de [2] e [3].

Combinamos todos esses argumentos e obtemos a prova.

References

[1] J. -M Coron, F. Marbach, F. Sueur, Small-time global exact controllability of the Navier-Stokes equation

with Navier slip-friction boundary conditions. J. European Mathematical Society, Electronically published on

February 11, 2020. doi: 10.4171/JEMS/952.

[2] E. Fernandez-Cara, M. Gonzalez-Burgos, S. Guerrero, J. -P. Puel, Null controllability of the heat

equation with boundary Fourier conditions: the linear case, ESAIM Control, Optimization and Calculus of

Variations, 12 (2006), no. 3, 442–465.

[3] S. Guerrero, Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-

slip boundary conditions, ESAIM: COCV 12 (2006), no. 3, 484–544.


HIERARCHICAL EXACT CONTROLLABILITY OF SEMILINEAR PARABOLIC EQUATIONS

WITH DISTRIBUTED AND BOUNDARY CONTROLS

F. D. ARARUNA1, E. FERNANDEZ-CARA2 & L. C. DA SILVA3

1Departamento de Matematica, UFPB, PB, Brasil, [email protected],2EDAN and IMUS, US, Sevilla, Espanha, [email protected],

3IFRN, RN, Brasil, [email protected]

Abstract

We present some exact controllability results for parabolic equations in the context of hierarchic control

using Stackelberg–Nash strategies. We analyze two cases: in the first one, the main control (the leader) acts

in the interior of the domain and the secondary controls (the followers) act on small parts of the boundary; in

the second one, we consider a leader acting on the boundary while the followers are of the distributed kind. In

both cases, for each leader an associated Nash equilibrium pair is found; then, we obtain a leader that leads the

system exactly to a prescribed (but arbitrary) trajectory. We consider linear and semilinear problems.

1 Introduction

Let Ω ⊂ RN (N ≥ 1) be a bounded domain with boundary Γ of class C2. Let O, O1, O2 ⊂ Ω be (small)

nonempty open sets and let S, S1 and S2 be nonempty open subsets of Γ. Given T > 0, we will set Q := Ω× (0, T )

and Σ := Γ × (0, T ). In this paper, 1A denotes the characteristic function of the set A.

We will consider parabolic systems of the formyt − ∆y + a(x, t)y = F (y) + f1O in Q,

y = v1ρ1 + v2ρ2 on Σ,

y(· , 0) = y0 in Ω

(1)

and pt − ∆p+ a(x, t)p = F (p) + u11O1

+ u21O2in Q,

p = gρ on Σ,

p(· , 0) = p0 in Ω,

(2)

where y0, p0, f , g, vi and ui are given in appropriate spaces, F : R → R is a locally Lipschitz-continuous function

and ρ, ρi ∈ C2(Γ), with

0 < ρ ≤ 1 on S, ρ = 0 on Γ \ S, 0 < ρi ≤ 1 on Si, ρi = 0 on Γ \ Si.

We will analyze the exact controllability to the trajectories of (1) and (2) following hierarchic control techniques,

as introduced by J.-L. Lions [1]. More precisely, we will apply the Stackelberg–Nash method, which combines

optimization techniques of the Stackelberg kind and non-cooperative Nash optimization techniques.

Let us define the secondary cost functionals for (1) and (2), respectively, as follows:

Ji(f ; v1, v2) :=αi

2

∫∫Oi,d×(0,T )

|y − ξi,d|2 dx dt+µi

2

∫∫Si×(0,T )

|vi|2 dσ dt, i = 1, 2, (3)

and

Ki(g;u1, u2) :=αi

2

∫∫Oi,d×(0,T )

|p− ζi,d|2 dx dt+µi

2

∫∫Oi×(0,T )

|ui|2 dx dt, i = 1, 2, (4)

229

230

where O1,d,O2,d ⊂ be nonempty open, ξi,d, ζi,d are given in L2(Oi,d × (0, T )), and αi, µi are positive constants.

Results based on Stackelberg–Nash strategies with one leader and several followers have been obtained in [3]

(resp. in [1, 2]) in the context of approximate (resp. exact) controllability. In all these papers, distributed controls

were considered.

The main goal in the present paper is to try to extend these results to systems of the kind (1) and (2), that is,

parabolic semilinear systems partially controlled from the boundary.

2 Main Results

Theorem 2.1. Suppose Oi,d ∩ O = ∅, i = 1, 2. Assume that one of the following conditions holds: either

O1,d = O2,d and ξ1,d = ξ2,d (5)

or

O1,d ∩ O = O2,d ∩ O. (6)

If the µi/αi (i = 1, 2) are large enough and F ∈ W 1,∞(R), there exists a positive function ς = ς(t) blowing up

at t = T with the following property: if y is a trajectory of (1) associated to the initial state y0 ∈ L2(Ω) and∫∫Oi,d×(0,T )

ς2|y − ξi,d|2 dx dt < +∞, i = 1, 2, (7)

then, for any y0 ∈ L2(Ω) there exist controls f ∈ L2(O × (0, T )) and associated Nash quasi-equilibria (v1, v2) such

that the corresponding solutions to (1) satisfy y(T, ·) = y(T, ·).

Theorem 2.2. Suppose that

S ⊂ Oi and Oi ∩ Oj,d = ∅, i, j = 1, 2. (8)

If the µi/αi are large enough and F ∈W 1,∞(R), there exists a positive function ς = ς(t) blowing up at t = T with the

following property: if p is a trajectory of (2) associated to the initial state p0 ∈ L2(Ω) and the ζi,d ∈ L2(Oi,d×(0, T ))

are such that ∫∫Oi,d×(0,T )

ς2|p− ζi,d|2 dx dt < +∞, i = 1, 2, (9)

then, for any p0 ∈ L2(Ω), there exist a control g ∈ H1/2,1/4(S × (0, T )) and an associated Nash quasi-equilibria

(u1, u2) such that the corresponding solution to (2) satisfies p(T, ·) = p(T, ·).

References

[1] araruna, f. d., fernandez-cara, e., guerrero, s., santos, m. c. New results on the Stackelberg-Nash

exact controllability of linear parabolic equations, Systems Control Lett., 104 (2017), 78–85.

[2] araruna, f. d., fernandez-cara, e., guerrero, s., santos, m. c. Stackelberg-Nash exact controllabilty

for linear and semilinear parabolic equations, ESAIM: Control Optim. Calc. Var., 21 (2015), 835–856.

[3] dıaz, j. i., lions, j. l. - On the approximate controllability of Stackelberg-Nash strategies, Ocean Circulation

and pollution Control: A Mathematical and Numerical Investigations, (Madrid, 1997), 17–27, Springer, Berlin,

2004.

[4] lions, j. l. - Hierarchic Control, Proc. Indian Acad. Sci. Math. Sci., 104 (4) (1994), 295–304.


INTERACAO ENTRE DISSIPACAO FRACIONARIA E NAO-LINEARIDADE DE MEMORIA NA

EXISTENCIA DE SOLUCOES PARA EQUACOES DE TIPO PLACA

LUIS GUSTAVO LONGEN1

1 Departamento de Matematica, UFSC, SC, Brasil, [email protected]

Abstract

Consideramos uma equacao de tipo de placas com inercia rotacional, sob os efeitos de um amortecimento

fracionario e uma nao-linearidade de tipo de memoria, isto e, consideramos o seguinte problema de Cauchyutt −∆utt −∆u+∆2u+ (−∆)θut =

∫ t

0

(t− s)−γ |u(s, ·)|p ds, (t, x) ∈ [0,∞)× Rn,

(u, ut)(0, x) = (u0(x), u1(x)), x ∈ Rn × Rn

(P)

onde θ ∈[0, 1

2

), γ ∈ (0, 1), p > 1. Nosso objetivo e determinar o expoente crıtico p que e limıtrofe entre a

existencia e a nao-existencia de solucoes globais para o problema dado, bem como entender como o amortecimento

fracionario interage com a nao-linearidade de memoria e como essa interacao pode interferir no valor de p. Com

esse fim, encontramos e utilizamos diversas estimativas de energia Lη − Lq com 1 ≤ η ≤ 2 ≤ q ≤ ∞, bem como

estimativas pontuais e de multiplicadores a Mikhlin-Hormander, (L1∩Lp)−Lp para p < 2, onde ha uma perda na

taxa de decaimento. Analisamos diversos cenarios baseados na dimensao n ≤ 4 e nos intervalos admissıveis para

os parametros θ e γ, que caracterizam o damping fracionario e a nao-linearidade de memoria, respectivamente.

A contraparte de resultados de nao-existencia e obtida utilizando uma tecnica de “funcoes teste modificadas”,

substituindo-se a condicao de suporte compacto por funcoes em C∞q (Rn), um espaco de funcoes infinitamente

diferenciaveis com decaimento polinomial no infinito. Esta modificacao se faz necessaria devido A nao-localidade

do operador Laplaciano Fracionario (−∆)θ.

1 Introducao

Buscamos determinar o expoente crıtico para (P). Nos resultados de existencia, para n ≤ 4, mostramos que p e

dado por uma competicao entre tres valores: um expoente de tipo de Fujita [1],

pc(n, γ, θ) := 1 +2 (1 + (1 − γ)(1 − θ))

(n− 2 + 2γ(1 − θ))+, (1)

um expoente γ−1 independente da dimensao [2], devido a forte influencia do termo de memoria nao-linear quando

γ 1, e um terceiro expoente, pc = 14−10θ7−2θ+2γ(1−θ) , fruto da perda de decaimento na regiao de alta frequencia que e

ocasionada pelo termo de inercia rotacional −∆utt.


Teorema 2.1 (Existencia de Solucoes Globais). Sejam n ∈ N, n ≤ 4, θ ∈[0, 12

), γ ∈ (0, 1), p > p e s = 2+2γ(1−θ),

com p := maxpc, γ

−1, pc. Entao, existe ε > 0 tal que, para dados iniciais

(u0, u1) ∈ A :=

(Hs(Rn) ∩ L1(Rn)

)×(Hs−1(Rn) ∩ L1(Rn)

), se p ≥ 2,(

Hs(Rn) ∩ L1(Rn) ∩ W 3,p(Rn))×(Hs−1(Rn) ∩ L1(Rn) ∩ W 2,p(Rn)

), se p < 2,

(1)

com ∥(u0, u1)∥A ≤ ε, existe uma solucao global para o problema (P), u ∈ C([0,∞), H2

)∩ C1

([0,∞), H1

).

231

232

Prova: A boa colocacao para o caso supercrıtico e mostrada com base solucao para o problema linear associado,

escrita como ulin(t, x) = K0(t, x) ∗ u0(x) + K1(t, x) ∗ u1(x). Utiliza-se o Princıpio de Duhamel para reescrever o

problema como

u(t, ·) = ulin(t, ·) +Gu(t, x) em Hk(Rn), (2)

onde

Gu(t, x) :=

∫ t

0

∫ τ

0

(τ − s)−γ(I − ∆)−1K1(t− τ, x) ∗ |u(s, x)|p ds dτ. (3)

Em seguida, define-se para T > 0 o espaco de Banach de evolucao X(T ) := C([0, T ], H2

)∩C1

([0, T ], H1

)e utiliza-se

estimativas para ulin e Gu em diversos espacos Hκ(Rn), de modo a provar que o operador G e bem-definido em

X(T ), leva bolas de X(T ) em bolas de X(T ), e e Lipschitz em X(T ). Com isso, pode-se aplicar o Princıpio de

Contracao de Picard e garantir a existencia e unicidade de solucao local para o problema dado. Por fim, como as

estimativas obtidas para u sao uniformes com relacao a T, toma-se o limite T ∞, obtendo-se assim a globalidade

das solucoes com relacao A variavel temporal.

Teorema 2.2 (Nao-Existencia de Solucoes Globais). Seja pc como em (1), defina p = maxpc, γ

−1

e fixe

q = n+ 2θ. Assuma que u0, u1 ∈ L1(⟨x⟩qdx) e ainda, que u1 satisfaca a seguinte condicao de sinal,∫Rn

u1 dx > 0.

Se existir uma solucao fraca (nao-trivial) global no tempo u ∈ Lp ([0,∞), Lp(Rn, ⟨x⟩−q)) para o problema (P), entao

p ≥ p.

Prova: Caso γ > n−2n : Suponha que u ≡ 0 seja uma solucao global para (P). Para T > 0 e R ≫ 1, definimos

φR(x) = ⟨R−1x⟩−q e ψT (t) = D1−γt|T

(ωT (t)β

), onde Dα

t|T denota o operador Derivada Fracionaria de ordem α de

Riemann-Liouville A direita, e ωT (t) := (1 − t/T )χ[0,T ]. Prova-se entao que para η > 0, a solucao u deve satisfazer∫ Rη

0

∫BR

ωT (t)β |u(t, x)|pφR(x) dt dx < Cα,εR−g(η)p′+n+η, (4)

onde g(η) e uma funcao auxiliar que satisfaz g(η)p′ − (n+ η) > 0 quando p < pc. Tomando-se o limite T,R ∞,

conclui-se u ≡ 0, uma contradicao.

References

[1] fujita, h. - On the blowing up of solutions of the Cauchy problem for ut = ∆u + u1+α., J. Fac. Sci. Univ.

Tokyo, p.109-124, 1966.

[2] cazenave, t., dickstein, f. and weissler f. - An equation whose Fujita critical exponent is not given by

scaling, Nonlinear Anal., 68, p.862-874, 2008.

[3] d’abbicco, m. and fujiwara, k. - A test function method for evolution equations with fractional powers of

the Laplace operator, Nonlinear Anal., 202, p.422-444, 2021.


EXPONENTES CRITICOS PARA UM SISTEMA PARABOLICO ACOPLADO COM

COEFICIENTES DEGENERADOS

RICARDO CASTILLO1, OMAR GUZMAN-REA2, MIGUEL LOAYZA3 & MARIA ZEGARRA4

1Departamento de Matematica, Universidad del Bıo Bıo, Concepcion, Chile, [email protected],2Departamento de Matematica, Universidade de Brasılia, Brasılia-DF, Brasil, [email protected],

3Departamento de Matematica, Universidade Federal de Pernambuco, Recife, Pernambuco, Brasil, [email protected],4Facultad de Ciencias Matematicas, Universidad Nacional Mayor de San Marcos, Lima, Peru, [email protected]

Abstract

Neste trabalho estudamos o seguinte sistema parabolico acoplado ut − div(ω(x)∇u) = trvp, vt −div(ω(x)∇v) = tsuq em RN × (0, T ), onde p, q > 0, com pq > 1; r, s > −1, as condicao inicial (u0, v0) ∈(L∞(RN ))2, com u0, v0 ≥ 0, e ω e uma funcao de classe Muckenhoupt A1+ 2

N. Estabelecemos resultados de

existencia local e global de solucoes nao negativas.

1 Introducao

Sejam T > 0 e N ≥ 0. Consideremos o seguinte sistema parabolico acopladout − div(ω(x)∇u) = trvp, RN × (0, T ),

vt − div(ω(x)∇v) = tsuq, RN × (0, T ),

u(0) = u0,RN ,

v(0) = v0,RN ,

(1)

onde (u0, v0) ∈ (L∞(RN ))2, com u0, v0 ≥ 0; p, q > 0, pq > 1; r, s > −1 e a funcao ω ou e

(A) ω(x) = |x|a com a ∈ [0, 1) se N = 1, 2 e a ∈ [0, 2N ) se N ≥ 3, ou

(B) ω(x) = |x|b com b ∈ [0, 1).

O problema (1) aparece em modelos termicos com difusao degenerada em um meio nao homogeneo e modelos

populacionais, veja [3, 4].

Solucoes para o problema (1) e entendida no seguinte sentido

Definicao 1.1. Sejam (u0, v0) ∈ (L∞(RN ))2, com u0, v0 ≥ 0, r, s > −1, pq > 1 e T ∈ (0,∞]. Entao chamamos

solucao do problema (1), se (u, v) ∈ (L∞(0, T ;L∞(RN )))2 e satisfaz

u(t) = S(t)u0 +

∫ t

0

S(t− σ)σrvp(σ)dσ

v(t) = S(t)v0 +

∫ t

0

S(t− σ)σsuq(σ)dσ,

(2)

para t > 0. Quando T = ∞, entao e dita solucao global no tempo. Onde S(t)z(x) =

∫RN

Γ(x, y, t)z(y)dy para t > 0,

e Γ(x, y, t) e a solucao fundamental do problema homogeneo ut − div(ω(x)∇u) = 0.

233

234


Neste trabalho apresentamos resultados que garantem a existencia local e global de solucoes nao negativas para o

problema (1). O resultado que garante a existencia local e o seguinte

Teorema 2.1. Assuma (A) ou (B), e sejam (u0, v0) ∈ [L∞(RN )]2, com u0, v0 ≥ 0. Entao, existe T > 0 tal que o

problema (1) possui uma unica solucao (u, v) definido sobre [0, T ] e satisfazem

sup0<t<T

(∥u(t)∥∞ + ∥v(t)∥∞) ≤ C0(∥u0∥∞ + ∥v0∥∞) (1)

Prova: Para demonstrar este resultado, constuımos uma sequencia e, em seguida, usamos as propriedades de

Γ(x, y, t) e adaptamos as ideias que aparecem em [2].

Teorema 2.2. Assuma que r, s > −1 e p, q ≥ 0, com pq > 1, e sejam γ1 = (r+1)+(s+1)ppq−1 , γ2 = (s+1)+(r+1)q

pq−1 ,

γ = maxγ1, γ2, r1∗ = N(2−α)γ1

, r2∗ = N(2−α)γ2

, onde α = a no caso (A) e α = b no caso (B).

(i) Se γ ≥ N2−α , entao o problema (1) nao tem solucoes globais nao triviais.

(ii) Se γ < N2−α , entao existem solucoes globais nao triviais para o problema (1).

Prova: Para demonstrar este resultado, utilizamos as propriedades de Γ(x, y, t) apresentadas em [2]. Logo,

utilizamos as ideas que aparecem em [1], para o operador do problema (1).

References

[1] castillo, r. and loayza, m. - Global existence and blowup for a coupled parabolic system with time-weighted

sources on a general domain. Z. Angew. Math. Phys., 70, 16, 2019.

[2] fujishima, y., kawakami, t. and sire, y. - Critical exponent for the global existence of solutions to a

semilinear heat equation with degenerate coefficients. Calc. Var. Partial Differential Equations, 58, 62, 2019.

[3] kamin, s. and rosenau, p. - Propagation of thermal waves in an inhomogeneous medium. Comm. Pure Appl.

Math., 34, 831-852, 1981.

[4] wang, w. and zhao, x.-q. - Basic Reproduction Numbers for Reaction-Diffusion Epidemic Models. SIAM J.

Applied Dynamical Systems, 11(4), 1652-1673, 2012.


SISTEMA DE BRESSE COM DISSIPACAO NAO-LINEAR NA FRONTEIRA

PATRICIA VILAR VITOR SALINAS1 & JUAN AMADEO SORIANO PALOMINO2

1Pma, UEM, PR, Brasil, [email protected],2 Dma, UEM, PR, Brasil, [email protected]

Abstract

Estudamos o sistema de Bresse com mecanismos de dissipacao nao-linear na fronteira. Utilizando conceitos e

resultados da Teoria de Semigrupos nao-lineares, provamos a existencia e unicidade de solucao para o sistema de

Bresse com mecanismos de dissipacao nao-linear na fronteira e mostramos a estabilidade exponencial do sistema

de Bresse sem qualquer condicao sobre as velocidades de propagacao das ondas.

1 Introducao

Considere o seguinte sistema de Bresse sem forcas externas

ρ1φtt − k(φx + ψ + lw)x − k0l(wx − lφ) = 0 em (0, L)× (0,∞),

ρ2ψtt − bψxx + k(φx + ψ + lw) = 0 em (0, L)× (0,∞),

ρ1wtt − k0(wx − lφ)x + kl(φx + ψ + lw) = 0 em (0, L)× (0,∞),

(1)

Um problema delicado no estudo do sistema de Bresse consiste em mostrar a estabilidade exponencial com

mecanismo de dissipacao na fronteira.

Nosso objetivo e novo no estudo de sistema de Bresse pois, trabalharmos com mecanismo de dissipacao na

fronteira nao-linear, o que e de grande relevancia na literatura. Primeiramente, foi obtido a existencia e unicidade

do sistema de Bresse sem forcas externas com as seguintes condicoes de fronteira

φ(0, t) = ψ(0, t) = w(0, t) = 0, ∀ t ≥ 0,

k(φx + ψ + lw)(L, t) + g1(φt(L, t)) = 0, ∀ t ≥ 0

bψx(L, t) + g2(ψt(L, t)) = 0, ∀ t ≥ 0

(2)

onde gi : R → R, para i = 1, 2, 3 sao termos dissipativos nao-lineares e condicoes iniciais

φ(·, 0) = φ0(·), φt(·, 0) = φ1(·)

ψ(·, 0) = ψ0(·), ψt(·, 0) = ψ1(·)

w(·, 0) = w0(·), wt(·, 0) = w1(·).

(3)

O escopo do nosso trabalho esta direcionado a estabilidade exponencial do sistema de Bresse sem forcas externas

com mecanismos de dissipacao nao-linear na fronteira agindo simultaneamente nas forcas axial e de cisalhamento e

no momento bending, sem a necessidade de velocidades iguais de propagacao de ondas e sem condicoes adicionais.

O trabalho que nos inspirou foi o de Lasiecka e Tatura [3], juntamente coma teoria de existencia para semigrupos

nao-lineares abordada nos trabalhos de [2] e [3].


Hipotese H-1: As funcoes nao lineares gi : R → R, para i = 1, 2, 3 satisfazem as seguintes condicoes :

235

236

(i) gi sao funcoes contınuas e crescentes sobre R;

(ii) gi(s)s > 0 para s = 0,

(iii) Existem m e M constantes tais que 0 < m < M e ms2 ≤ gi(s)s ≤Ms2, |s| > 1.

Teorema 2.1. Assumindo a Hipotese H-1, temos que para cada Y0 ∈ D(A) existe uma unica solucao forte para

(1). Alem disso, se Y0 ∈ H entao (1), possui uma unica solucao generalizada.

Diversos trabalhos que envolvem sistema de Bresse tem como hipotese uma condicao puramente matematica.

Esta condicao e a ou diferenca entre as velocidades de propagacoes de ondas, a saber,

ρ1ρ2

=k

be k = k0. (4)

Mostramos que a energia associado a uma solucao do sistema de Bresse com dissipacoes nao lineares na fronteira

decai exponencialmente sem a hipotese (4). Desssa forma, trabalhamos com sistemas fisicamente possıveis.

Com multiplicadores convenientes nos obtivemos a seguinte desigualdade:∫ T

0

E(t)dt ≤ C

[∫ T

0

[ρ1(φt)

2(L) + ρ2(ψt)2(L) + ρ1(wt)

2(L) + (g1(φt(L)))2 + (g2(ψt(L)))2

+(g3(wt(L)))2]Ldt+

∫ T

0

∫ L

0

(φ2 + ψ2 + w2)dxdt+ E(T )

] (5)

Lemma 2.1. Para T suficientemente grande, existe uma constante C > 0 tal que∫ T

0

∫ L

0

(φ2 + ψ2 + w2)dxdt ≤ C

∫ T

0

[(g1(φt(L)))2 + (g2(ψt(L)))2 + (g3(wt(L)))2

+ ρ1(φt)2(L) + ρ2(ψt)

2(L) + ρ1(wt)2(L)]Ldt,

(6)

para toda solucao forte U = (φ,ψ,w, Φ,Ψ,W ) do sistema de Bresse dado em (1)-(3).

Da desigualdade dada em (5) e do lema anterior, obtemos o resultado:

Teorema 2.2. Seja T > 0 suficientemente grande. Entao, a energia do sistema dado por (1)-(3) satisfaz

E(T ) ≤ CT

∫ T

0

[(ρ1φ

2t + ρ2ψ

2t + ρ1w

2t + (g1(φt))

2 + (g2(ψt))2 + (g3(wt))

2)

(L)]Ldt. (7)

Utilizando as funcoes definidas por Lasieka e Tataru e procedendo de maneira analoga como em [3], tendo em

vista o Teorema (2.2), a solucao do sistema (1)-(3) satisfaz o Teorema 2 de [3].

References

[1] andrade, j. - Controlabilidade exata a zero na fronteira para o sistema de Bresse e controlabilidade interna

para o sistema de Bresse termoelastico, Tese de Doutorado, Programa de Pos-graduacao em Matematica,

Universidade Estadual de Maringa, Maringa, 2017.

[2] barbu, V. - Nonlinear semigroup and differencial equations in Banach spaces, Sditura Academici RomA¢ne,

Bucuresti, 1974.

[3] brezis. H. Functional Analysis, Sobolev Spaces and Partial Differential Equations., Springer, 2010.

[4] lasieka, I. and tataru, D. - Uniform boundary stabilization of semilinear wave equations with nonlinear

boundary damping. Differential and integral Equations, v. 6, n. 3, p. 507-533, 1993.


SOBRE A CONTROLABILIDADE UNIFORME DOS SISTEMAS BURGERS-α NAO-VISCOSO E

VISCOSO

RAUL K. C. ARAUJO1, ENRIQUE FERNANDEZ-CARA2 & DIEGO ARAUJO DE SOUZA3

1Departamento de Matematica, UFPE, PE, Brasil, [email protected],2Departamento E.D.A.N, Universidade de Sevilha, Sevilha, Espanha, [email protected],

3Departamento E.D.A.N, Universidade de Sevilha, Sevilha, Espanha, [email protected]

Abstract

Analisamos neste trabalho a controlabilidade global de certas famılias de EDP’s chamadas de sistemas

Burgers-α nao-viscoso e viscoso. Nessas equacoes, o termo convectivo da famosa equacao de Burgers e substituıdo

por um termo regularizado, o qual e induzido por um filtro de Helmholtz de comprimento de onda caracterıstico

α. Provamos primeiramente um resultado de controlabilidade global exata (uniforme em relacao a α) para o

sistema Burgers-α nao-viscoso usando, principalmente, o metodo do retorno e um argumento de ponto-fixo. Apos

isso, a controlabilidade global exata e uniforme a estados constantes e deduzido para o sistema viscoso. Para tal

proposito, provamos primeiramente um resultado de controlabilidade local exata e, feito isso, estabelecemos um

resultado de controlabilidade global aproximada para estados inicial e final regulares.

1 Introducao

Sejam L, T > 0 dados. Neste trabalho, consideramos as seguintes duas famılias de sistemas controlados:

yt + zyx = p(t) em (0, T ) × (0, L),

z − α2zxx = y em (0, T ) × (0, L),

z(·, 0) = vl, z(·, L) = vr em (0, T ),

y(·, 0) = vl em Il,

y(·, L) = vr em Ir,

y(0 , ·) = y0 em (0, L),

(1)

onde Il = t ∈ (0, T ) : vl(t) > 0 e Ir = t ∈ (0, T ) : vr(t) < 0 e

yt − µyxx + zyx = p(t) em (0, T ) × (0, L),

z − α2zxx = y em (0, T ) × (0, L),

z(·, 0) = y(·, 0) = vl em (0, T ),

z(·, L) = y(·, L) = vr em (0, T ),

y(0 , ·) = y0 em (0, L).

(2)

Os sistemas (1) e (2) sao chamados, respectivamente, de sistemas Burgers-α nao-viscoso e viscoso. Devemos

destacar tambem que a terna (p, vl, vr) e o par (y, z) representam, respectivamente, os controles e os estados

associados. O parametro µ > 0 e a viscosidade do fluido e α > 0 e o comprimento de onda caracterıstico do

chamado filtro de Helmholtz.

237

238


Teorema 2.1. Seja α > 0 dado. O sistema Burgers-α nao-viscoso (1) e globalmente exatamente controlavel em

C1. Mais precisamente, dados y0, y1 ∈ C1([0, L]), existe um controle fonte pα ∈ C0([0, T ]), um par de controles de

fronteira (vαl , vαr ) ∈ C1([0, T ];R2) e um par de estados associados (yα, zα) ∈ C1([0, T ] × [0, L];R2) satisfazendo (1)

e

yα(T, ·) = yT in (0, L).

Alem disso, existe uma constante C > 0 (dependendo de L, T , y0 e yT , mas independente de α), tal que

∥(zα, yα)∥C1([0,T ]×[0,L];R2) + ∥pα∥C0([0,T ]) + ∥(vαl , vαr )∥C1([0,T ];R2) ≤ C.

Prova: Conforme vemos em [1], a prova baseia-se, essencialmente, em dois argumentos principais: metodo do

retorno e argumento de ponto fixo. A aplicacao do metodo do retorno consistiu em linearizar o sistema nao-linear

(1) ao redor de uma trajetoria apropriada e provar que o sistema linearizado assim obtido e globalmente controlavel

a zero. Apos isso, efetuamos uma ligeira perturbacao nesse sistema linearizado e provamos, usando um argumento

de ponto fixo de Banach, que o sistema perturbado e localmente controlavel a zero. O resultado do teorema segue

facilmente daı.

Teorema 2.2. Seja α > 0 dado. Entao, o sistema Burgers-α viscoso e globalmente exatamente controlavel,

em L∞, a trajetorias constantes. Noutras palavras, para quaisquer y0 ∈ L∞(0, L) e N ∈ R, existe um controle

fonte pα ∈ C0([0, T ]), um par de controles de fronteira (vαl , vαr ) ∈ H3/4(0, T ;R2) e um par de estados associados

(yα, zα) ∈ L2(0, T ;H1(0, L;R2)) ∩ L∞(0, T ;L∞(0, L;R2)), satisfazendo (2),

yα(T, ·) = N in (0, L),

e a seguinte estimativa

∥pα∥C0([0,T ] + ∥(vαl , vαr )∥H3/4(0,T ;R2) ≤ C,

onde C > 0 e uma constante que depende de L, T , y0 e N , mas independe de α. Alem disso, se y0 ∈ H10 (0, L)

entao a mesma conclusao ocorre com

(yα, zα) ∈ L2(0, T ;H2(0, L;R2)) ∩H1(0, T ;L2(0, L;R2)).

Prova: Conforme vemos em [1], a prova divide-se em tres etapas: efeito regularizante, controlabiliade aproximada

para dados regulares e controlabilidade local exata a trajetorias de classe C1([0, T ].

References

[1] Araujo, R. K. C.; Fernandez-Cara, E.; Souza, D. A.. On the uniform controllability for a family of

non-viscous and viscous Burgers-α systems, artigo aceito para publicacao em ESAIM : Control, Optimisation

and Calculus of Variations.

[2] Cheskidov A., Holm, D., Olson, E. and Titi, E.. On a Leray-α model of turbulence, Proc. R. Soc. A,

461, 629–649, 2005.

[3] Holm, D. and Staley, M.. Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM

J. Appl. Dyn. Syst. 2, 323–380, 2003.


SISTEMA DE BRESSE COM ACOPLAMENTO TERMOELASTICO NO MOMENTO FLETOR E

LEI DE FOURIER

ROMARIO TOMILHERO FRIAS1 & MICHELE DE OLIVEIRA ALVES2

1IMECC, UNICAMP, SP, Brasil, [email protected],2Departamento de Matematica, UEL, PR, Brasil, [email protected]

Abstract

Neste trabalho estuda-se um sistema de Bresse com acoplamento termoelastico no momento fletor

considerando a lei de Fourier para o fluxo de calor. O principal objetivo e fazer uma apresentacao mais detalhada

da existencia, unicidade e comportamento assintotico do problema descrito em [1]. A teoria de semigrupos de

operadores lineares e utilizada para garantir a existencia e unicidade de solucao. Uma condicao necessaria e

suficiente e dada para a obtencao da estabilidade exponencial do semigrupo e verifica-se que sob certas condicoes

obtem-se decaimento polinomial da solucao.

1 Introducao

O objetivo do presente trabalho e apresentar resultados descritos em [1] de uma forma didatica e detalhada. Alem

disso, utilizando um resultado obtido em [2], melhoramos resultados estabelecidos em [1], no que concerne as taxas

de decaimento polinomial, para o sistema termoelastico de Bresse.

O sistema estudado e dado por

ρ1φtt − k(φx + ψ + lw)x − k0l(wx − lφ) = 0 em (0,∞) × (0, L), (1)

ρ2ψtt − bψxx + k(φx + ψ + lw) + γθx = 0 em (0,∞) × (0, L), (2)

ρ1wtt − k0(wx − lφ)x + kl(φx + ψ + lw) = 0 em (0,∞) × (0, L), (3)

ρ3θt − αθxx + γψxt = 0 em (0,∞) × (0, L). (4)

Considerando as seguintes condicoes iniciais:

φ(0, ·) = φ0, φt(0, ·) = φ1, ψ(0, ·) = ψ0, ψt(0, ·) = ψ1,

w(0, ·) = w0, wt(0, ·) = w1, θ(0, ·) = θ0.(5)

E condicoes de fronteira de Dirichlet

φ(t, 0) = φ(t, L) = ψ(t, 0) = ψ(t, L) = w(t, 0) = w(t, L) = 0,

θ(t, 0) = θ(t, L) = 0 para t ∈ (0,∞), (6)

ou condicoes de fronteira de Dirichlet-Neumann

φ(t, 0) = φ(t, L) = ψx(t, 0) = ψx(t, L) = wx(t, 0) = wx(t, L) = 0,

θ(t, 0) = θ(t, L) = 0 para t ∈ (0,∞). (7)

Onde os coeficientes ρ1, k, ρ2, ρ3 = γm , α = k1γ

m , b, γ, l, k0, k1 e m sao constantes positivas e as funcoes φ,ψ,w e θ

descrevem, respectivamente, a oscilacao vertical, o angulo de rotacao da secao transversal, a oscilacao longitudinal

e a variacao de temperatura de uma viga fina, arqueada e com comprimento L.

239

240

O sistema (1)-(7) foi estudado por [1], onde os autores mostraram que a estabilidade da solucao do sistema esta

diretamente ligada as seguintes constantes

χ :=

∣∣∣∣1 − k

k0

∣∣∣∣ e χ0 :=

∣∣∣∣ bk ρ1 − ρ2

∣∣∣∣. (8)

Em [1], Fatori e Rivera mostraram que a solucao do sistema e exponencialmente estavel se, e somente se, χ = χ0 = 0.

Alem disso, mostraram que se χ = χ0 = 0 nao se satisfaz o semigrupo associado ao sistema (1)-(5) em geral possui

uma taxa de decaimento t−1/6 e que para o caso em que χ = 0 e χ0 = 0 a taxa e t−1/3.

Obtemos neste trabalho uma melhora em relacao ao artigo apresentado em [1], a saber, mostraremos que para o

caso χ0 = 0 e χ = 0 a taxa de decaimento da solucao do semigrupo associado ao sistema (1)-(7) pode ser melhorada

para t−1/2, e que no caso de χ = 0 e o semigrupo associado ao sistema (1)-(5) com condicoes de fronteira (7) ser

polinomialmente estavel, a taxa de decaimento da solucao nao pode ser melhor que t−1/2.


Teorema 2.1.

ρ1ρ2

= k

bou k = k0, (1)

entao o semigrupo associado ao sistema (1)-(5) com condicoes de fronteira (7) nao e exponencialmente estavel.

Prova: Para obter esta prova veja [3].

Teorema 2.2. Suponha que ρ1, ρ2, ρ3, b, k, k0, α, γ > 0, χ0 = 0 e χ = 0. Entao, existe uma constante C > 0,

independentes do dado inicial U0 ∈ H, tal que

||U(t)||H ≤ C

t1/2||U0||D(Ai); t→ ∞. (2)


Teorema 2.3. Se χ = 0 e o semigrupo associado ao sistema (1)-(5) com condicoes de fronteira (7) for

polinomialmente estavel, entao a taxa de decaimento do semigrupo nao pode ser melhor que

||U(t)||H2≤ C

t1/2||U0||D(A2); t→ ∞. (3)


References

[1] L. H. FATORI and J. E. M. RIVERA - Rates of decay to weak thermoelastic Bresse system. IMA Journal

of Applied Mathematics, 1–24, 2010.

[2] MORAES, G. E. B. and SILVA, M. A. J. - Arched beams of Bresse type: observability and application

in thermoelasticity. NONLINEAR DYNAMICS, v. 103, 2365–2390, 2021.

[3] FRIAS, R. T. - Sistema de Bresse com acoplamento termoelastico no momento fletor e lei de Fourier.

Dissertacao de mestrado, UEL., 2020.


LINEABILITY OF MULTILINEAR SUMMING OPERATORS

LINDINES COLETA1

1Departamento de Matematica, UFPB, PB, Brasil, [email protected]

Abstract

We present a result on the recent notion of directed/geometric lineability, introduced by Favaro, Pellegrino

and Tomaz (2020), related to the class of multilinear Λ-summing operators. Some applications are obtained, in

particular, we prove that the set of m-linear operators on Banach spaces with values on ℓp that are absolutely

but not multiple summing is (1, c)−spaceable. This is a joint work with N. G. Albuquerque and D. Tomaz.

1 Introduction

Let E1, . . . , Em be Banach spaces over K, the complex or real scalar field. The Bohnenblust-Hille multilinear

inequality [1], provides that every m−linear form E1 × · · · ×Em → K is multiple ( 2mm+1 ; 1)-summing and, moreover,

2mm+1 is optimal. The Defant-Voigt theorem (see [2]) also tells us that every multilinear form E1 × · · · ×Em → K is

(r; 1)-summing, for r ≥ 1. Combining these results we concluded that

Πabs(r;1)(E1, . . . , Em,K) \ Πmult

(r;1)(E1, . . . , Em,K) = ∅

whenever 1 ≤ r < 2mm+1 . Here Πabs

(p;q) denotes the class of absolutely (p, q)-summing operators and Πmult(p;q) the class

of multiple (p, q)-summing operators. Therefore is natural to investigate the lineability (and also spaceability) of

the set of absolutely but not multiple summing multilinear operators. We deal with theses problems in the general

concept of multilinear Λ-summing operators (see [1] and [3]), providing results in the more restrictive variant of

lineability/spaceability notion recently presented in [4]. Next we present a precise definition of these concepts. As

usual, the topological dual and the closed unit ball of a Banach space E will be denoted by E′ and BE , respectively,

c stands for the cardinality of R and m will always be a positive integer.

Definition 1.1. Let E1, . . . , Em, F Banach spaces, r := (r1, . . . , rm), s = (s1, . . . , sm) ∈ [1,+∞)m and Λ ⊂ Nm a

set of indexes, a m-linear operator T : E1 × · · · × Em → F is Λ-(r, s)-summing, if there is a constant C > 0 such

that N∑i1=1

· · ·

(N∑

im=1

∥∥∥T (x(1)i1, . . . , x

(m)im

)1Λ(i1, . . . , im)

∥∥∥rmF

) rm−1rm

· · ·

r1r2

1r1

≤ C

m∏k=1

supϕk∈BE′

k

(N∑i=1

∣∣∣ϕ(x(k)i )∣∣∣pk

) 1pk

,

for all N ∈ N and x(k)i ∈ Ek, k = 1, . . . ,m, i = 1, . . . , N , where 1Λ is the characteristic function of Λ.

The set of operators that fulfill the previous inequality is denoted by ΠΛ(r,s) (E1, . . . , Em;F ), which is a Banach

space endowed with the usual norm taken as the infimum of the constants C > 0. Notice that when Λ = Nm and

Λ = (i, . . . , i) : i ∈ N, the class of multiple, absolutely summing operators is recovered, respectively. It is worth

pointing out that the concepts of summing operators can be investigated on quasi-Banach spaces which topological

dual is nontrivial (see [5]).

241

242

Definition 1.2. Let α, β, λ be cardinal numbers and V be a vector space, with dimV = λ and α < β ≤ λ. A

set A ⊂ V is (α, β)-lineable (respec. (α, β)-spaceable), if it is α-lineable and for every subspace Wα ⊂ V , with

Wα ⊂ A ∪ 0 and dimWα = α, there is a subspace (respec. closed subspace) Wβ ⊂ V , with dimWβ = β and

Wα ⊂Wβ ⊂ A ∪ 0.

Observe that this definition encompass and refine the original lineability notion when α = 0.

2 Main Results

Let Λ ⊂ Λ∗ subsets of Nm, E =: E1 × · · · × Em, with E1, . . . , Em Banach spaces and p ∈ (0,∞). Our main result

provides that the set∏Λ

(r,s)(E; ℓp)\∏Λ∗

(r,s)(E; ℓp) is either empty or (1, c)−spaceable. Among others applications, we

provide the spaceability of the class of absolutely but not multiple summing operators and also the class of linear

operators that fails to be absolutely summing.

Theorem 2.1. Let E1, . . . , Em be Banach spaces, E =: E1 × · · · × Em, p ∈ (0,∞), r := (r1, . . . , rm), s :=

(s1, . . . , sm) ∈ [1,+∞)m and Λ ⊂ Λ∗ ⊂ Nm sets of indexes. Let us consider the spaces of m−linear summing

operators ΠΛ(r,s)(E; ℓp) and ΠΛ∗

(r,s)(E; ℓp). Then

ΠΛ(r,s)(E; ℓp) \ ΠΛ∗

(r,s)(E; ℓp)

is either nonempty or (1, c)-spaceable.

Corollary 2.1. Let r := (r1, . . . , rm), s := (s1, . . . , sm) ∈ [1,+∞)m and p ∈ (0,+∞). Then

Πabs(r,s) (E1, . . . , Em; ℓp) \ Πmult

(r,s) (E1, . . . , Em; ℓp)

is either empty or (1, c)-spaceable.

It is well known that, for 0 < p < 1, the identity I : ℓp → ℓp is a non-(r, s)−absolutely summming operator for

any 1 ≤ s ≤ r < ∞. Hence, the set L(ℓp, ℓp) \ Π(r,s)(ℓp, ℓp) is not empty. Using this fact and a direct application

of the technique used in Theorem 2.1, we obtain the following result.

Proposition 2.1. Let 0 < p < 1 and let 1 ≤ s ≤ r <∞. Then L (ℓp; ℓp)\⋃

1≤s≤r<∞

∏(r,s) (ℓp; ℓp) is (1, c)−spaceable.

In the next result we deal with multilinear operators with values in ℓp(Γ).

Proposition 2.2. Under the same assumptions of the Theorem 2.1, if the set∏Λ

(r,s)(E; ℓp) \∏Λ∗

(r,s)(E; ℓp) is non-

empty, it is (α, card(Γ))-lineable for all α < card(Γ).

References

[1] albuquerque, n. a. g. et al. - On summability of multilinear operators and applications, Ann. Funct. Anal.

9, 574-590, 2018.

[2] araujo, g., pellegrino, d. - Optimal estimates for summing multilinear operators, Linear and Multilinear

Algebra, 65, 930-942, 2017.

[3] botelho, g., freitas, d. - Summing multilinear operators by blocks: the isotropic and anisotropic cases, J.

Math. Anal. Appl. 490, 2020.

[4] favaro, v. v., pellegrino, d. m. and tomaz, d. - Lineability and spaceability: a new approach, Bull. Braz.

Math. Soc. (N.S), 51, 27-46, 2020.

[5] maddox, i. j. - A non-absolutely summing operators, J. Austral. Math. Soc. (Series A), 43, 70-73, 1987.


POLINOMIOS HOMOGENEOS NAO ANALITICOS E UMA APLICACAO AS SERIES DE

DIRICHLET

MIKAELA A. OLIVEIRA1

1ICE, UFAM, AM, Brasil, mado11318@gmailcom

Abstract

Na dissertacao estuda-se polinomios homogeneos contınuos que nao sao analıticos. Os principais resultados

referem-se a existencia de estruturas lineares constituıdas por polinomios nao analıticos e, tambem, uma aplicacao

desses polinomios as series de Dirichlet. Com esse fim, comecamos com o estudo dos polinomios homogeneos

entre espacos de Banach e suas principais propriedades. Em seguida, sao exibidas as construcoes do polinomio

2-homogeneo dada por Toeplitz e do polinomio m-homogeneo, m ≥ 2, devida a Bohnenblust e Hille. Com o

auxılio desses polinomios e gerado um subespaco vetorial isomorfo ao espaco ℓ1, gozando da propriedade de

que os seus elementos (nao nulos) sao polinomios homogeneos que nao sao analıticos num determinado vetor.

Em particular, o conjunto dos polinomios homogeneos nao analıticos em c0 e espacavel. Por fim, como uma

aplicacao exibimos a solucao do Problema de Convergencia Absoluta de Bohr, que consiste na determinacao da

distancia maxima entre as abscissas de convergencia absoluta e uniforme de uma serie de Dirichlet, tendo como

ferramenta util em sua solucao o polinomio de Bohnenblust e Hille.

1 Introducao e resultados principais

e sabido dos cursos introdutorios de analise complexa que uma funcao de uma variavel complexa e holomorfa se e

somente se e analıtica, i.e., pode ser representada localmente como uma serie infinita de monomios de uma variavel.

Para funcoes de finitas variaveis complexas e possıvel mostrar que esse resultado continua valido. Por muito tempo

se acreditou que para funcoes em infinitas variaveis isso tambem valeria.

Em 1913 o matematico Toeplitz exibiu um exemplo de uma funcao holomorfa em que sua representacao por

serie de potencias nao convergia em todo ponto. Mais precisamente ele construiu um polinomio 2-homogeneo em c0

que nao era analıtico em todos os pontos do seu domınio. Denotando P(2c0) o espaco dos polinomios 2-homogeneos

contınuos em c0, Toeplitz mostrou o seguinte resultado

Teorema 1.1. Existe P ∈ P(2c0) de modo que para cada ε > 0, existe z ∈ ℓ4+ε P tal que nao e analıtico.

Posteriormente, Bohnenblust e Hille em [1] para resolverem um problema na serie de Dirichlet estenderam a

construcao de Toeplitz a polinomios m-homogeneos (m ≥ 2) e mostraram o seguinte

Proposicao 1.1. Para cada m ≥ 2 fixo, existe P ∈ P(mc0) tal que para cada ε > 0, existe z ∈ ℓ 2mm−1+ε de modo

que P nao e analıtico.

O trabalho de Bohnenblust e Hille [1] tem sido bastante explorado nos ultimos anos por ter implicacoes no

estudo da analiticidade de polinomios m-homogeneos contınuos. Em 2016 J. Alberto Conejero, Juan B. Seoane-

Sepulveda e Pablo Sevilla Peris com base nos polinomios de Bohnenblust e Hille mostraram em [2] que o conjunto

dos polinomios m-homogeneos contınuos nao analıticos em c0, (que denotaremos por Nm) e espacavel em P(mc0),

ou seja, Nm ∪ 0 contem um espaco vetorial fechado de dimensao infinita.

Teorema 1.2. Para cada m ≥ 2, o conjunto Nm ∪ 0 contem uma copia isomorfa de ℓ1. Em particular Nm e

espacavel em P(mc0).

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244

Como aplicacao estudamos o Problema de convergencia absoluta de Bohr que consiste na determinacao da largura

maxima da faixa em que uma serie de Dirichlet converge uniformemente mas nao absolutamente. Este problema foi

considerado por Harold Bohr em 1913 enquanto investigava a distancia maxima entre as abscissas de convergencia

de uma serie de Dirichlet. Bohr consideriou o numero

S = supσa(D) − σu(D) : D e uma serie de Dirichlet

onde σa(D) e σu(D) denotam as abscissas de convergencia absoluta e uniforme de uma serie de Dirichlet D,

respectivamente. Bohr mostrou que S ≤ 1

2, entretanto ele nao conseguiu nenhum exemplo de modo que

σa(D) − σu(D) =1

2.

Apesar de nao ter resolvido este problema, Bohr forneceu ferramentas que levaram a sua solucao. Ele percebeu que

as series de Dirichlet e as series de potencias formais estavam relacionadas por meio dos numeros primos. Dada uma

serie de Dirichlet D(s) =∑ann

−s considere para cada n ∈ N sua decomposicao em numeros primos n = pα11 · · · pαn

n .

Pela unicidade dessa decomposicao cada n corresponde um unico α = (α1, . . . , αN ). Entao, definindo cα = apα cada

serie de Dirichlet corresponde uma unica serie de potencias∑cαz

α, onde zα = zα1 · · · zαN . Essa correspondencia

e chamada de Transformada de Bohr e permite traduzir problemas sobre series de Dirichlet em termos de series de

potencias.

O problema de convergencia absoluta de Bohr foi resolvido apenas em 1931 por Hille e Bohnenblust, e uma das

ferramentas usadas foi o polinomio m-homogeneo que tinham construıdo. Com isso eles mostraram em [2] que para

cada m ∈ N, existem series de Dirichlet tais que

σa − σu =m− 1

2m.

Isto implicava no seguinte resultado

Proposicao 1.2. Temos

S =1

2,

e o supremo e atingido, ou seja, existe uma serie de Dirichlet tal que σu(D) = 0 e σa(D) =1

2.

References

[1] bohnenblust, h. f. and hille e. - On the absolute convergence of Dirichlet series., Ann. of Math. 32,

600-622 (1931).

[2] conejero, j. a., seoane-sepulveda, j. b. and sevilla-peris, p. - Isomorphic copies of ℓ1 for m-

homogeneous non-analytic Bohnenblust-Hille polynomials. Mathematische Nachrichten, 218-225, 2016.

[3] defant, a., garcia, d., maestre, m. and sevilla-peris, p. - Dirichlet series and holomorphic functions

in high dimensions, Cambridge University Press, 2019.


IDEAIS INJETIVOS DE POLINOMIOS HOMOGENEOS ENTRE ESPACOS DE BANACH

GERALDO BOTELHO1 & PEDRO C. BAZAN2

1Faculdade de Matematica, UFU, MG, Brasil, [email protected],2 Faculdade de Matematica, UFU, MG, Brasil, [email protected]

Abstract

Neste trabalho estudamos os ideais injetivos de polinomios homogeneos, com enfase na envoltoria injetiva.

Posteriormente, apresentamos a descricao da envoltoria injetiva de um ideal de composicao e aplicacoes desta

descricao sao fornecidas.

1 Introducao

As nocoes de ideal injetivo e envoltoria injetiva aparecem inicialmente para ideais de operadores lineares (veja [3]),

e posteriormente sao generalizados de forma natural para ideais de polinomios homogeneos. Os ideais injetivos

sao importantes por possuir estreita relacao com as injecoes metricas (ou isometrias lineares) e com a restricao de

contradomınio de um operador. Sao importantes tambem porque muitos ideais interessantes de operadores e de

polinomios sao injetivos. Este trabalho e baseado nos resultados principais de [2], artigo no qual os ideais injetivos

de polinomios foram primeiramente estudados.


Ao longo deste trabalho, m denota um numero natural qualquer e as letras E,F,G e H denotam espacos de Banach

quaisquer, reais ou complexos. Por E′ denotamos o dual topologico do espaco E, por L(E;F ) denotamos o espaco

dos operadores lineares de E em F e por P(mE;F ) o espaco dos polinomios m-homogeneos contınuos de E em F .

Para comodidade do leitor, apresentamos a definicao de ideal de polinomios.

Definicao 2.1. Um ideal de polinomios (homogeneos) e uma subclasse Q da classe de todos os polinomios

homogeneos contınuos entre espacos de Banach tal que suas componentes Q(mE;F ) = P(mE;F )∩Q, onde m ∈ Ne E e F sao espacos de Banach arbitrarios, satisfazem as seguintes condicoes:

(1) Q(mE;F ) e um subespaco vetorial de P(mE;F ) que contem os polinomios m-homogeneos de tipo finito.

(2) Se u ∈ L(G;E), P ∈ Q(mE;F ) e t ∈ L(F ;H), entao t P u ∈ Q(mG;H).

Se ∥ · ∥Q : Q −→ R e uma funcao tal que (Q(mE;F ), ∥ · ∥Q) e um espaco normado (de Banach) para quaisquer

espacos de Banach E e F e numero natural m, e satisfaz as seguintes condicoes:

(I) ∥idm : K −→ K , idm(λ) = λm∥Q = 1 para todo m ∈ N, e

(II) Se u ∈ L(G;E), P ∈ Q(mE;F ) e t ∈ L(F ;H), entao ∥t P u∥Q ≤ ∥t∥ · ∥P∥Q · ∥u∥m,entao (Q, ∥ · ∥Q) e chamado de ideal normado (de Banach) de polinomios.

Dado um ideal de polinomios Q, por Qm denotamos a sua componente m-linear, isto e, Qm(E;F ) := Q(mE;F )

para todos E e F espacos de Banach. Qm e chamado tambem de ideal de polinomios m-homogeneos. e claro que

Q1 e um ideal de operadores lineares. Para a teoria basica de ideais de operadores, veja [3].

Uma injecao metrica e um operador linear j : E −→ F tal que ∥j(x)∥ = ∥x∥ para todo x ∈ E. Por

IE : E −→ ℓ∞(BE′) denotamos a injecao metrica canonica (veja [3, C.3.3]).

245

246

Definicao 2.2. (i) Dizemos que um ideal de polinomios Q e injetivo se dados P ∈ P(mE;F ) e uma injecao metrica

j : F −→ G tais que j P ∈ Q(mE;G), tem-se que P ∈ Q(mE;F ).

(ii) Um ideal normado de polinomios (Q, ∥ · ∥Q) e injetivo se Q e um ideal injetivo de polinomios e, na situacao

acima, ∥P∥Q = ∥j P∥Q.

A seguir veremos as propriedades principais da envoltoria injetiva de um ideal de polinomios.

Proposicao 2.1. Seja (Q, ∥ · ∥Q) um ideal normado de polinomios. Entao existe um (unico) menor ideal normado

injetivo de polinomios (Qinj, ∥ · ∥Qinj) que contem Q e tal que ∥ · ∥Qinj ≤ ∥ · ∥Q. Para P ∈ P(mE;F ),

P ∈ Qinj(mE;F ) ⇐⇒ IF P ∈ Q(mE; ℓ∞(BF ′)) e ∥P∥Qinj := ∥IF P∥Q.

Mais ainda, o ideal (Qinj, ∥ · ∥Qinj) e de Banach se o ideal (Q, ∥ · ∥Q) for de Banach. O ideal Qinj (ideal normado

(Qinj, ∥ · ∥Qinj)) e chamado de envoltoria injetiva do ideal Q (ideal normado (Q, ∥ · ∥Q)).

Corolario 2.1. (a) Um ideal de polinomios Q e injetivo se, e somente se, Q = Qinj.

(b) Um ideal normado de polinomios (Q, ∥ · ∥Q) e injetivo se, e somente se, Q = Qinj e ∥ · ∥Q = ∥ · ∥Qinj .

Os conceitos e propriedades de ideais injetivos de operadores (veja [3, 4.6]) sao naturalmente recuperados do

caso polinomial ao se considerar o caso linear m = 1 no que foi apresentado acima. Neste caso, denotamos tambem

por I inj a envoltoria injetiva de um ideal de operadores I. Analogamente, obtemos a definicao e as propriedades

de ideal injetivo de polinomios m-homogeneos ao considerarmos m fixo no que foi apresentado acima.

Seja I um ideal de operadores. Um polinomio P ∈ P(mE;F ) pertence a I P(mE;F ) se existem um espaco

de Banach G, um polinomio Q ∈ P(mE;G) e um operador linear u ∈ I(G;F ) tais que P = u Q. O ideal de

polinomios I P e chamado de ideal de composicao. Para maiores informacoes sobre esse ideal, veja [1]. O resultado

a seguir descreve a envoltoria injetiva de um ideal de composicao.

Teorema 2.1. Seja I um ideal de operadores. Entao (I P)inj = I inj P.

Muitas consequencias decorrem da formula acima. Vejamos algumas.

Corolario 2.2. As seguintes afirmacoes sao equivalentes para um ideal de operadores I:(a) I e injetivo.

(b) I P e um ideal injetivo de polinomios.

(c) (I P)m e um ideal injetivo de polinomios m-homogeneos para algum m ∈ N.

Por A denotamos o ideal dos operadores lineares que podem ser aproximados, na norma usual, por operadores

lineares de posto finito, por K denotamos o ideal dos operadores lineares compactos, por PA denotamos o ideal

dos polinomios que podem ser aproximados, na norma usual, por polinomios de posto finito e por PK o ideal dos

polinomios compactos. De [3, 4.6.13] sabemos que Ainj = K. Usando o Teorema 2.1 consegue-se a versao polinomial

desse resultado:

Corolario 2.3. (PA)inj = PK.

References

[1] botelho, g.; pellegrino, d. and rueda, p. - On composition ideals of multilinear operators and

homogeneous polynomials. Publ. Res. Inst. Math. Sci., 43, 1139–1155, 2007.

[2] botelho, g. and torres, l. a. - Injective polynomial ideals and the domination property. Results Math.,

75, Paper No. 24, 12 pp., 2020.

[3] pietsch, a. - Operator Ideals, North-Holland, 1980.


EXISTENCE OF POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEMS WITH

P -LAPLACIAN OPERATOR

FRANCISCO J. TORRES1

1Departamento de Matematica - Universidad de Atacama - Copiapo - Chile

Abstract

This paper is concerned with the existence of positive solutions for three point boundary value problems of

Riemann-Liouville fractional differential equations with p-Laplacian operator. By means of the properties of the

Green’s function and Avery-Peterson fixed point theorem, we establish a condition ensuring the existence of at

least three positive solutions for the problem.

1 Introduction

This paper investigates the existence of at least three positive solutions for the following nonlinear fractional

boundary value problem, (FBVP in short),

Dβ0+(φp(Dα

0+u(t))) + a(t)f(t, u, u′) = 0, for each t ∈ [0, 1],

Dα0+u(0) = u(0) = u′(0) = 0, Dα−2

0+ u(0) = Dα−20+ u(1) = γu(η),

where η ∈ (0, 1), γ ∈(

0, Γ(α−1)ηα−2

), φp(s) = |s|p−2s, p > 1, Dα

0+ , Dβ0+ are the Riemann-Liouville fractional derivatives

with α ∈ (3, 4] and β ∈ (0, 1].

To establish the existence of multiple positive solutions of FBVP, throughout this paper, we assume that f and a

satisfy the following conditions:

(H1) f ∈ C([0, 1] × [0,∞) × [0,∞), [0,∞)) is a given nonlinear function.

(H2) a ∈ L∞[0, 1] and there exists m > 0 such that a(t) ≥ m a.e. t ∈ [0, 1].

2 Main Results

In this section we deduce the existence of at least three positive solutions of the FBVP by using the well know

Avery-Peterson fixed point theorema; see [1].

Let γ and θ be nonnegative continuous convex functionals on P , ω be nonnegative continuous concave functional

on P and ψ be a nonnegative continuous functional en P . Then for positive numbers a, b, c and d, we define the

following convex sets:

P (γ, d) = x ∈ P : γ(x) < d,

P (γ, ω, b, d) = x ∈ P : b ≤ ω(x), γ(x) ≤ d,

P (γ, θ, ω, b, c, d) = x ∈ P : b ≤ ω(x), θ(x) ≤ c, γ(x) ≤ d,

and a closed set

R(γ, ψ, a, d) = x ∈ P : a ≤ ψ(x), γ(x) ≤ d.

247

248

Theorem 2.1. [1] Let P be a cone in Banach space E. Let γ, θ be nonnegative continuous convex functionals

on P , ω be a nonnegative continuous concave functional on P , and ψ be a nonnegative continuous functional on

P satisfying ψ(λx) ≤ λψ(x) for 0 ≤ λ ≤ 1, such that for some positive numbers M and d, ω(x) ≤ ψ(x) and

∥x∥ ≤Mγ(x) for x ∈ P (γ, d).

Suppose that T : P (γ, d) → P (γ, d) is completely continuous and there exist positive numbers a, b, c with a b = ∅ and ω(Tx) > b for

x ∈ P (γ, θ, ω, b, c, d);

(S2) ω(Tx) > b for x ∈ P (γ, ω, b, d) with θ(Tx) > c;

(S3) 0 ∈ R(γ, ψ, a, d) and ψ(Tx) < a for x ∈ R(γ, ψ, a, d) with ψ(x) = a.

Then T has at least three fixed points x1, x2, x3 ∈ P (γ, d) such that γ(xi) ≤ d, i = 1, 2, 3; ω(x1) > b; ψ(x2) > a,

ω(x2) < b; ψ(x3) < a.

Now, for convenience, we denote

r1 =

(∥a∥∞

Γ(β + 1)

)1−q NB(2, β(q − 1) + 1)

,

r2 =

(m

Γ(β + 1)

)1−q

τ1−α

[∫ 1

τ

G(1, s)sβ(q−1)ds+γ

N

∫ 1

0

G(η, s)sβ(q−1)ds

]−1

,

r3 =

(∥a∥∞

Γ(β + 1)

)1−qΓ(α) − (α− 1)γηα−2

B(2, β(q − 1) + 1),

where N = Γ(α− 1) − γηα−2.

Theorem 2.2. Suppose that (H1 −H2) hold and there exist constants 0 < a φp(r2b), (t, u, u′) ∈ [τ, 1] × [b, bτ1−α] × [0, d],

(A3) f(t, u, u′) < φp(r3a), (t, u, u′) ∈ [0, 1] × [τα−1a, a] × [0, d].

Then the FBVP has at least three positive solutions u1, u2 and u3 satisfying

max0≤t≤1

u′i(t) ≤ d, i = 1, 2, 3; minτ≤t≤1

u1(t) > b;

max0≤t≤1

u2(t) ≥ a with minτ≤t≤1

u2(t) < b; and max0≤t≤1

u3(t) < a.

References

[1] Avery, R. I., Peterson, A. C. - Three positive fixed points of nonlinear operators on ordered Banach

Spaces, Comput. Math. Appl., 42, 313–322, 2001.

[2] Podlubny, I. - Fractional Differential Equation, Academic Press, San Diego, 1999.


ESQUEMAS DE DIFERENCAS FINITAS PARA SECAO CIRCULAR

TATIANA DANELON DE ASSIS1 & SANDRO RODRIGUES MAZORCHE2

1Programa de Pos-graduacao em Matematica, UFJF, MG, Brasil, [email protected],2Instituto de Ciencias Exatas, UFJF, MG, Brasil, [email protected]

Abstract

Neste trabalho apresenta-se um esquema de diferencas finitas em coordenadas polares para o laplaciano de

uma funcao com condicao de contorno de Dirichlet e, a partir dele, e desenvolvido um esquema semelhante para a

norma do gardiente. A principal dificuldade e tratar a singularidade na origem que surge devido A mudanca para

o sistema de coordenadas polares. Alem disso, sao montadas matrizes auxiliares para facilitar a implementacao

numerica dessa aproximacao. Uma aplicacao dos esquemas desenvolvidos e feita utilizando um modelo de torcao

elastoplastica para avaliar a qualidade dos resultados e discutir sobre sua importancia.

1 Introducao

O tema deste trabalho foi motivado por estudos acerca do problema da torcao elastoplastica (PTE), que consiste

em definir regioes de plasticidade formadas na secao transversal Ω ⊂ R2 de uma barra submetida a torcao. A

solucao desse problema – descrito detalhadamente em [3] – satisfaz a seguinte desigualdade variacional:

u ∈ K∇ :

∫Ω

∇u · ∇(v − u) dxdy ≥ −τ∫Ω

∇(v − u) dxdy, ∀v ∈ K∇, (1)

com condicao de contorno de Dirichlet u = 0 em ∂Ω. O conjunto K∇ =v ∈ H1

0(Ω) : ∥∇v∥ ≤ 1

define os

deslocamentos admissıveis e τ e uma constante fısica. A resolucao numerica do problema envolve a discretizacao

via metodo das diferencas finitas do laplaciano e da norma do gradiente de u. Assim, surgiram dificuldades para

alguns formatos de barra, como secoes em L (abordada em [1]) e circulares. Embora a utilizacao de diferencas

finitas com coordenadas polares seja antiga, ha detalhes que precisam ser tratados com atencao.

Seja o disco de raio R definido por Ω =

(x, y) : x2 + y2 < R

, aplica-se a mudanca de coordenadas x =

r cos θ, y = r sin θ, na qual r =√x2 + y2 e θ = arctan (y/x). Para Ωp = (r, θ) : 0 < r < R, 0 ≤ θ < 2π, tem-se::

∆u(r, θ) =∂2u

∂r2+

1

r

∂u

∂r+

1

r2∂u2

∂θ2, ||∇u(r, θ)||2 =

(∂u

∂r

)2

+1

r2

(∂u

∂θ

)2

. (2)

Note que surge uma singularidade na origem nas equacoes (2). Diferentes estrategias podem ser adotadas para

trazer a regularidade desejada, como o metodo de deslocamento da grade descrito em [2]. Neste caso, a malha de

diferencas finitas e definida de forma que o eixo radial seja composto por semi-inteiros, ou seja, ri = (i− 1/2)hr e

θj = (j − 1)hθ, sendo hr = R/(Nr + 1/2) e hθ = 2π/Nθ, para i = 1, 2, ..., Nr + 1, j = 1, 2, ..., Nθ + 1.

Utilizando diferencas centradas para aproximar o laplaciano, para i = 2, 3, ..., Nr e j = 1, 2, ..., Nθ, tem-se:

∆u(ri, θj) ≈Ui+1,j − 2Ui,j + Ui−1,j

h2r+

1

ri

Ui+1,j − Ui−1,j

2hr+

1

r2i

Ui,j+1 − 2Ui,j + Ui,j−1

h2θ, (3)

onde Ui,j denota a solucao aproximada de u(ri, θj). Quanto aos pontos de fronteira, segue da condicao de contorno

e da periodicidade do disco que UNr+1,j = 0 e Ui,Nθ+1 = Ui,1. Para i = 1 o termo U0,j se anula na equacao (3),

pois r1 = hr/2. Logo esse metodo permite que o laplaciano seja resolvido sem nenhuma condicao de polo.

249

250

O objetivo principal deste trabalho e desenvolver um esquema de diferencas finitas semelhante para a norma do

gradiente, ja que os resultados encontrados dizem respeito apenas ao laplaciano. Aplicando diferencas centradas:

||u(ri, θj)||2 ≈U2i+1,j − 2Ui+1,jUi−1,j + U2

i−1,j

4h2r+

1

r2i

U2i,j+1 − 2Ui,j+1Ui,j−1 + U2

i,j−1

4h2θ, (4)

para i = 2, 3, ..., Nr e j = 1, 2, ..., Nθ. A aproximacao para os pontos de fronteira e analoga ao caso anterior, mas para

i = 1 o termo U0,j nao se anula. Para contornar essa questao, optou-se por diferencas progressivas para aproximar

U1,j . Para implementar numericamente, duas matrizes A e B foram montadas tais que ||u||2 ≈ (AU)2 + (BU)2,

sendo U = [U1 U2 ... UNr+1]t e U i = [Ui,1 Ui,2 ... Ui,Nθ+1]t. Sao elas:

A =

−2L 2L

−L 0 L. . .

. . .. . .

−L 0 L

−L 0 0

−2L 0

, B =

M

. . .

M

0

, (5)

nas quais:

L =

1/(2hr)

. . .

1/(2hr)

1/(2hr) 0

, M =

−1/hθ 1/hθ

−1/(2hθ) 0 1/(2hθ). . .

. . .. . .

−1/(2hθ) 0 1/(2hθ)

1/(2hθ) −1/(2hθ) 0 0

1/hθ 1/hθ 0

. (6)


O PTE foi resolvido numericamente atraves de dois modelos, via I. complementaridade (utilizando ∆u) e II.

minimizacao (utilizando ||∇u||). O caso circular com τ constante possui solucao analıtica, entao foi possıvel calcular

o erro. Para (I) o erro relativo medio foi de 0, 0554% e para (II) foi de 0, 8010%. O esquema de complementaridade

teve melhor desempenho, mas no geral pode-se dizer que ambos apresentaram resultados satisfatorios.

A equivalencia entre as regioes plasticas definidas por ||∇u|| = 1 ou |u| = d foi um marco na area, sendo

d : Ω → R a funcao que mede a menor distancia de cada ponto ate a fronteira. Esse resultado configura o PTE

como um problema tipo obstaculo, com u sendo representado por uma membrana e d o obstaculo que restringe

seu deslocamento. Assim, e possıvel encontrar as regioes sem envolver o gradiente. Essa equivalencia, entretanto,

e valida para o PTE classico, mas existem outras variacoes de τ compatıveis com a formulacao matematica que

fogem ao significado fısico do problema. Nesses casos, ha o surgimento de novas regioes (ou o alargamento de regioes

existentes) em que a norma do gradiente iguala ou supera a unidade sem que haja contato com o obstaculo. Por

isso um esquema em diferencas finitas para ||∇u|| e importante, ampliando o campo de aplicacoes matematicas.

References

[1] danelon, t. a. - Resolucao numerica do modelo da torcao elastoplastica via complementaridade mista para

secao em L. Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 8, in press.

[2] lai, m. c. - A note on finite difference discretizations for Poisson equation on a disk. Numerical Methods for

Partial Differential Equations, 17, 199-203, 2001.

[3] rodrigues, j. f. - Obstacle Problems in Mathematical Physics, Mathematical Studies, North-Holland, 1987.

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